Statement and Assumptions

This is a very important topic. In recent exams 5 questions of this type are asked. So mastering this chapter is essential for success in competitive exams.
FORMAT OF THE PROBLEM
Directions : In each question below a statement (or a passage) is followed by two assumptions numbered I and II. An assumption is something supposed or taken in for granted. You have to consider the statement and the following assumptions and then decide which of the assumptions is implicit in the statement.
 Give answer:
(a) if only assumption I is implicit.
(b) if only assumption II is implicit.
(c) if either assumption I or assumption II is implicit.
(d) if neither of the assumptions is implicit.
(e) if both the assumptions are implicit.

Example:
Statement: Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I. Study material prepared by XYZ is of good quality.
II. Getting a high score in the exam is desirable.

ASSUMPTION: An assumption is something which is assumed, supposed and taken for granted.
When someone says something he may not put every aspect of his idea into words. That which is left unsaid or taken for granted is called an assumption.
IMPLICATION: Implication is something which is implied. It is the hidden meaning of the statement. Sometimes implications are also taken as assumptions.
DIFFERENCE BETWEEN ASSUMPTION AND IMPLICATION
An assumption is something on which the statement is based, while implication is something which is derived from and therefore based upon the statement.While doing problems on statement assumption
one should read each statement very carefully and whenever find anything presupposed or taken for granted, immediately notice it. Misreading of some facts leads to incorrect answer. So while reading one should be conscious about words because sometimes a single word's presence or absence changes
the entire meaning of a sentence. Points to be remembered while evaluating assumptions :
The use of definite words may lend a different tone to a statement. So one should be careful about them. For example some key words like 'only', 'best', 'strongest', 'all', 'definitely', 'certainly' etc. impact a kind of exclusiveness to the sentence and thus reduce its range.
Example:
Statement : You should use computers to increase the efficiency of your office.
Assumptions: I. Only the use of computers can increase the efficiency of office.
                       II. The use of computers can increase the efficiency of the office.
Assumption II is a valid one but I is not valid. In the statement it is suggested that computers may be used to increase the efficiency but it is not said that only computers can increase the efficiency. It is only one of the methods by which the efficiency can be increased.
1) When two suggestions / events / facts, A and B are connected by the conjuction,
a) A because / as a result of B. Assumption - B leads to A
b) A therefore / hence B. Assumption - A leads to B
c) A even after / inspite of / despite B. Assumption - Usually A does not occur, when B occurs.
Ex: The house was robbed inspite of the tight security in that area.
2) Whenever you come across a connotative or connotive phrases note that,
a) "It is true that" can be expressed as
i) It would be correct to say that....
ii) Even the most sceptic men would agree that......
iii) It can be claimed with reasonable degree of truth that.....
b) It is false can be expressed as
i) It is highly misleading to say that.....
ii) Nothing could be farther from the truth that....
iii) It is baseless to say that.....
3) If the statement tells about the existence of something the assumption should be that it does exists. If it is said that something is absent, the assumption should be that it not exists.
Statement :The increase in the improved number of educated women.
Assumption : There are educated women in the society.
4) If an adjective is attached (unconditionally) to any subject, it must be assumed that the subject, does have the quality.
Ex. Statement : The talkative nature of Sheela attracted everybody's attention.
   Valid assumption: Sheela is talkative.
5) Suppose a fact / report / observation / study / data 'A' followed by a suggested course of action B is given. If some negative aspect of A is mentioned,The valid assumptions can be
i) A needs improvement.
ii) The negative aspects of A are undesirable
iii) B will improve A
iv) The advantages of adopting B far outweigh the disadvantages of net adapting it.
6) Advertisements / notices / appeals Statements in the form of an advertisement or an official notice or a notice issued in the public interest or an appeal are often asked in the recent exams.The following assumptions are valid in these cases.
i) An advertisement / appeal / notice does have some effect.
ii) In the case of an advertisement, that which is being highlighted is looked for and expected by the people.
iii) In the case of a public - interest notice, it is the duty of those who issue it, to issue such notices.
iv) In case of a public interest notice, what is being advised must be beneficial for people and its non-practice harmful in some way.
v) In case of an appeal, the reason for issuing it exists.
vi) In case of an official notice, the effect of its implementation will be beneficial for the organisation.

TIPS FOR QUICK ANSWER
From the explanations given above you can find that there may be more than one valid assumption for a statement. In almost all cases they are not mutually exclusive. So if you find that both the assumptions are valid mark the answer 'both are implicit'. The answer choice "Either I or II is implicit" is to be marked only, when one assumption excludes the other. This case occurs only very rarely. So be careful before marking this choice.
In questions with more than two assumptions its combinations like I & II are implicit, II & III are implicit, I & III are implicit etc. are given as answer choices. If you are sure that assumption III is not implicit you can eliminate answer choices which contains III thus reducing the number of answer choices. Then select answer from the remaining choices. You can keep in mind some typical
examples which will help you while answering questions in the examination. If the given assumption is just contrary to the given statement or if it is not connected with the given statement then the assumption can be immediately rejected. In some cases the assumption is a restatement of the given
statement. The candidate should be careful in evaluating this type of assumptions.
Eg Statement : Small things are beautiful.
Invalid assumptions:
I. Big things are not beautiful.
II. Big things are ugly.
III. Small things are not beautiful.
Assumption III is just the contrary of what is given in the statement. More saying of small things are beautiful does not mean that big things are not beautiful or ugly. So assumptions I & II are not implicit.
Directions : In each question below is given a statement followed by two assumptions - numbered I and II. An assumption is something supposed or taken for granted. You have to consider the statement and the following assumptions and decide which of the assumptions is implicit in the statement.
Give answer (a) if only assumption I is implicit;
(b) If only assumption II is implicit;
(c) If either I or II is implicit;
(d) if neither I nor II is implicit and
(e) if both I and II are implicit.
Solved Examples:
1. Statement : Read the study material prepared by XYZ to get a high score in the exam.
Assumptions :
I Study material prepared by XYZ is of good quality.
II Getting a high score in the exams is desirable.
Ans (e). Both the assumptions are implicit in the statement. It is said in the statement that the study material prepared by XYZ helps to get a high score, which means that the study material by XYZ is of good quality thus assumption I is implicit. Reading the study material is recommended to get a high score in the exam, that is getting a high score, in the exams is desirable. Hence II is also implicit.

2. Statement : This book is invested to guide the layman to study tailoring in the absence of a teacher.
Assumptions :
I. A teacher of tailoring may not be available to everyone.
II. Tailoring can be learn with the help of a book.
Ans(e). Both the assumptions are implicit. The book is intended to teach in the absence of a teacher. From this we can enter that the absence of teacher is a possibility, thus assumption I is valid. Since the book is intended to teach tailoring, assumption II is also implicit.

3. Statement : Buy pure and natural pearls of company `X' - an advertisement.
Assumption :
I. Artificial pearls can be prepared.
II. People do not mind paying more for pure and natural pearls.
Ans (a). The advertisement tells that the pearls of company X is natural, this means that artificial pearls may be available in the market. So assumption I is implicit. But the advertisement tells nothing about the price of the pearls. So II cannot be implicit

Data Sufficiency

To begin understanding Data Sufficiency, it’s helpful to look at the official directions and answer choices for these problems, courtesy of the Graduate Management Admissions Council.
Directions
This Data Sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of
mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question asked;
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
You should see from the answer choices that the name of the game is “is the data sufficient?”, so let’s spend some time discussing what constitutes sufficient information.

More important than those solutions, however, are these takeaways:
• A statement is sufficient when it guarantees exactly one (and only one) answer to the question.
• This means that in a Yes/No question, you have sufficient information if the answer is “Definitely Yes” or if the answer is “Definitely No”. You do not have sufficient information when the answer is “Sometimes Yes but Sometimes No” (or “Maybe”).
• This means that in a “What is the Value?” question, you have sufficient information when you can pin down exactly one value for the question, but you do not have sufficient information when more than one value is possible.
• Data Sufficiency questions require attention to detail – the drills on the previous page came in pairs, and to the untrained eye each pair might have seemed the same. But subtle differences in what was given or asked – variable squared vs. variable cubed; cube vs. rectangular box; 2A + 3P vs. 3A + 2P – can make all the difference.
As tricky as Data Sufficiency can look at first, the way in which they are constructed ensures that you can attack them systematically. As you know, the answer choices are always the same. And you should also know that there are only two types of question stems that can ask:
1. Yes or No Questions
• A statement gives multiple solutions, but they all give the same answer.
• A statement provides a no answer instead of a yes answer.
2. What Is the Value? Questions
• A statement appears to be giving one value because you have assumed properties of the number that were not actually given (positive, integers, etc.).
• Restrictions were placed in the problem that you did not properly leverage (for instance, the problem is asking for the number of children, which must be an integer and cannot be negative). 

 Avoid assumptions. Every time you approach a Data Sufficiency problem, you must actively consider any assumptions that you may have been baited into making. Avoiding assumptions is perhaps the most important skill in all with Data Sufficiency.

1. Yes or No Question
Statement:Is x > 3?
Arguments:(1) The sum of x and the square of x is 12.
                   (2) square of x > 9
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution. A
Question Type: Yes/No. This question asks: “Is x > 3?”
Given information in question stem or diagram: No important information is given in the question stem.
Statement 1: The first step in this statement is to translate the wording into the following equation: x + x2 = 12. Since this is a quadratic equation, you should set everything equal to zero so that x2 + x – 12 = 0. Factoring this, you see that (x + 4)(x - 3) = 0 and x would be -4 or 3. The difficulty in this statement is that many people assume that this information is not sufficient because there are two values, one negative and one positive. However, remember that to prove sufficiency in a yes or no question requires only a definitive answer, not one value. Since each of these values (-4 and 3) gives a “no” answer to the question, this statement is sufficient. The answer is either A or D.
Statement 2: x2 > 9. If x2 > 9 then either x > 3, which gives you a “yes” answer, or x < 3, which gives you a “no” answer. For example x could be -5 (which when squared is > 9) or 5 (which when squared is also > 9). This statement is thus not sufficient, and the correct answer is A.
Note: This question is created to prey on two common mistakes, one relating to Data Sufficiency itself and one relating to algebra:
 1.) People (even those who have done lots of data sufficiency) tend to forget to look for the “no” answer inYes/No questions and they often make mistakes about what is really requiredfor sufficiency on Yes/No questions.
 2.) People forget about the negative possibilities when dealing with squared variables in inequalities.

2. What Is the Value? Question
Statement: If a, b, and c are distinct positive integers where a < b < c and √abc = c, what is
the value of a?
Arguments:(1) c = 8
                   (2) The average of a, b, and c is 14/3
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed
Solution:D
Question Type: What Is the Value? This question asks for the specific value of
positive integer a.
Given information in the question stem or diagram: There is a lot of information SOLUTIONS to leverage from this question stem. a, b, and c are distinct positive integers; a< b < c; and the square root of abc = c. You should first manipulate the last one algebraically by squaring both sides to see that abc = c2. Divide both sides by c (you can do this because you know that c cannot be 0 from the question stem) and the equation becomes ab = c. So before you even go to the statements you know that ab = c and all of the variables are different positive integers.Statement 1: c = 8. Combined with what you learned from the question stem, this means that ab = 8. Since a and b are distinct positive integers and a < b, the only possibility is a = 2 and b= 4. You might consider a = 1 and b = 8 but since
the integers must be distinct, you cannot have b = 8 since c = 8. This is sufficient but you will only see that if you properly leverage every piece of information given in the question stem. Remember: When you are given even a small piece of information in the question stem it is usually very important. The correct answer is A or D.
Statement 2: The average of a, b, and c is 14/3
. This means that the total of a+ b + c = 14. This statement is even trickier than the last but requires a similar leveraging of all available information. It may seem at first glance that there are many possibilities for the values of a, b, and c. However, the only way that ab = c and a + b + c = 14 is for a = 2, b= 4, and c = 8. There is no other way to have three distinct numbers add up to 14 and have ab = c. This statement is also sufficient and the correct answer is D. This question provides an excellent example of a phenomenon you will see often in Data Sufficiency: When a lot of information
is given in the question stem, statements are usually sufficient with much less information than you might first think.

Syllogism

                       The word "syllogism" is given by Greeks which means 'inference' or 'deduction'. It was introduced by Aristotle.An example of a question of syllogism is given below.
Directions : In the following questions, two statements are given followed by two conclusions. You have to study the two statements and then decide which of the conclusions follow from the statements. Mark the right answer from (1), (2), (3), (4) and (5)

Statements : All plants are trees.
                    No trees are green.
Conclusions : I. Some plants are green.
                      II. No plants are green.
1) Only I follows
2) Only II follows
3) Both I and II follow
4) Either I or II follows
5) Neither I nor II follows
This is a typical question of syllogism. Here the choice (2) is true. Later on we can discuss
the method to reach at the answer choice. Now let us see some definitions related to syllogism.

PROPOSITION
A proposition is a sentence that makes a statement and gives a relation between two or more terms.
In logic, any statement is termed a proposition.
Eg : i) All windows are rods
      ii) No cloth is a bay
     iii) Some students are members
     iv) Some green are not white
The parts of proposition are given below.
i) Subject : A subject is the part of the proposition about which something is being said.
ii) Predicate : Predicate is the part of the proposition denoting that which is affirmed
or denied about the subject.

eg : In the proposition All novels are songs, something is being said about novels. So novels is the subject. Songs is the predicate here because it affirmed about the subject.

CLASSIFICATION OF PROPOSITIONS
i) Universal positive proposition: A proposition of the form All S are P is called a universal positive proposition. A universal positive proposition is denoted by A.
eg : All girls are disciplined.
      All bulbs are lions.
i) Universal negative proposition :A proposition of the form No S is P is called a universal negative proposition. It is usually denoted by E.
eg : No professors is lazy.
       No boxes are baskets.
ii) Particular positive Proposition : A proposition of the form Some S are P is called a particular positive proposition. It is usually denoted by I.
eg : Some boys are smarts.
       Some boys are cats.
iv) Particular negative proposition : A proposition of the form Some S are not P is called particular negative proposition. It is denoted by the letter O.
eg : Some flowers are not grapes.
       Some fans are not black.
In syllogism, there are two types of inferences.
1) Mediate inference : Here conclusion is drawn from two propositions. For example, if you are given All cats are dogs and All dogs are animals, then a conclusion of the form All cats are animals could
be drawn from it.
2) Immediate inference :
Here conclusion is drawn from only one given proposition. For example if a given statement is All gates are blue, then based on this a conclusion could be drawn that Some blue are gates. This is a case of immediate inference.
Two important cases of immediate inference is given below.
a) Implications : If a given proposition is A - type, then it also implies that the I - type conclusion must be true. Let us verify it by considering the proposition, All elephants are big. This statement
naturally implies that the conclusion Some elephants are big must be true. Similarly we
can prove that an E - type proposition also implies an O - type conclusion.
b) Conversion
 Two steps are to be followed in conversion. The first step is to change the subject as the
predicate and the predicate as the subject. The second step is to change the type of the given
proposition to the pattern given in the following table.
Type of the given proposition      Type of the proposition after conversion
                 A                                                          I
                 E                                                          E
                 I                                                            I
                O                                           Cannot be converted

Let us consider the statement Some posters are good looking. This can be converted by using the above table as Some good looking are posters. In the same way, No books are pencils can be converted as No pencils are books.

HIDDEN PROPOSITION
You may find it difficult to categorise some propositions of the form Rahim is brilliant, Every man talks English, Not a single student passed the exam, No student except Prem was present,etc. We shall know, how to find the hidden propositions in such sentences.

A - type hidden propositions :
• All positive propositions beginning with 'each', 'every' and 'any'.
• A positive sentence with a particular person as its subject.
• A positive sentence with a very definite exception.
eg : Each of them plays football.
       He should be awarded.
All members except Kavitha have a share of profit.

E - type hidden proposition
• All negative sentences beginning with 'no one', 'none' and 'not a single'
• A sentence with a particular person as its subject but a negative sense.
• A negative sentence with a very definite exception.
• An interrogative sentence which is used to make an assertion.
eg : None can escape from death.
       Swathi is not an IAS officer.
No student except Salim has attend the party. Is there any person who can cheat himself?

I - type hidden propositions :
• Positive propositions beginning with words such as 'most', 'a few', 'mostly', 'generally',
'almost', `frequently', and negative propositions beginning with words such
as 'few', 'seldom', `hardly', 'scarcely','rarely' and 'little'.
• A positive sentence with an exception which is not definite.
eg : Very few writers research before they write.
      Seldom are people not jealous.
    All students except five have failed.

O - type hidden propositions :
• All negative propositions beginning with words such as 'all', 'every', 'any' and 'each'.
• Negative propositions with words as 'most', 'a few', 'mostly', 'generally', 'almost', and `frequently'.
• Positive words beginning with 'few', 'seldom', 'hardly', scarcely', 'rarely' and little.
• A negative sentence with an exception which is not definite..
e.g. : All men are not honest
         Most of the books have not been read.
         Girls are usually not brave.
         Rarely is a rich man worried.
         No students except a few are absent.

EXCLUSIVE PROPOSITIONS
A statement beginning with'only', 'alone','none but' or 'none else but' is called exclusive proposition. Such propositions can be reduced to A or E or I type.
Only brave men are pilots.
This sentence means that "No coward man is a pilot" and "All pilots are brave men".

SOLUTION OF SYLLOGISM BY ANALYTICAL METHOD
There are two steps to be followed for solving syllogism by analytical method. A problem of syllogism consists of two propositions which have one common term. This common term will be the predicate of the first proposition and the subject of the second. If this condition is not satisfied in the given propositions, they should be aligned accordingly.
eg : Statement : All birds are trees.
                         Some trees are cows.
Here the common term is 'trees'. Also it satisfies the above said condition. Hence the statements are properly aligned.
Let us consider another example.
eg : Statement : All pencils are bottles
                          All bricks are pencils.
Here the common term is 'pencil'. But it does not satisfy the given condition. So we have to align this pair. This can be aligned easily by changing the order of the statements. The aligned pair will be
All bricks are pencils.
All pencils are bottles.

eg : Statements : No watch is hat
                          All pins are hats.
In this pair, the common term is 'hat' and it is the predicate of both the sentences. So we have to align the sentences by converting any of the sentences and changing the order if needed. After alignment, the above example will become
All pins are hats
No hat is watch.
While aligning a given pair of statements, the priority should be given while converting, to I- type statements to E-type statements and then to A - type statement, in that order. That is, the rule of IEA should be followed.
After aligning the given pair of statements, the conclusion can be easily drawn by using the following table.
Statement - I      Statement - II        Conclusion
          A        +            A              =         A
          A        +            E              =         E       
          A        +            E              =        O*
          E        +            I               =        O*
           I        +            A             =       I
           I        +            E             =        O
No definite conclusion can be drawn for other combinations like A+I, O+A etc, which are not mentioned in the above table.
For the above given combinations which are aligned properly, the conclusion is a proposition whose subject is the subject of the first statement and whose predicate is the predicate of the second statements. The common terms disappears.
In the above table, O* implies that the conclusion is of type - O, whose subject is the predicate of the second statement and the predicate of the conclusion is the subject of the first statement.

SOLVED EXAMPLES.
1. Statements :All bags are toys.
                        All toys are keys.
The sentences are already aligned. From the above given Table, A+A=A. Hence the conclusion is of type - A whose subject is the subject of the first proposition and the predicate is the predicate of the second proposition. So the conclusion is All bags are keys.
2. Statements : All teachers are readers.
                       All teachers are writers.
This pair is not properly aligned because the subject of both the sentences is 'teachers'. Since both the sentences are of type - A, we may convert any of them. So the aligned pair is Some readers are teachers.
All teachers are writers.
Here the conclusion will be of type - I because I+A=I. The conclusion is Some readers are writers.
3. Statements : Some chocolates are toffees.
                        All chocolates are pastries.
The subject of both the sentences is the same.By the rule of IEA, we convert the I - type statement.
So the aligned pair is, Some toffees are chocolates.All chocolates are pastries I+A=I. So the conclusion is Some toffees are pastries.
4. Statements :All lights are balls
                        No bats are lights
By changing the order of the statements itself we can align the sentences. The aligned pair is
No bats are lights. All lights are balls.E+A=O*. So the conclusion is,
Some balls are not bats.
5. Statements :Some caps are red.
                        No clip is red.
Here the common term is 'red' which is the predicate of both the sentences. By the rule of IEA, we convert the I - type statement. After conversion, the given pair becomes,
Some red are caps.
No clip is red.
Now by changing the order of the statements, we can align the sentences. So the aligned pair is,
No clip is red.
Some red are caps.
The conclusion is of type O* since E+I=O*. Hence the conclusion is Some caps are not clips.
6. Statements : Some powders are not soaps.
                          All soaps are detergents.
The given pair is properly aligned. But no definite conclusion can be drawn from this type
because it is a O+A - type combination.

IMMEDIATE INFERENCE
Now let us consider an example which has two statements as well as two conclusions.
eg.  Statements: All novels are stories.
       All stories are songs.
Conclusion : (i) All novels are songs.
                    (ii) Some songs are novels.
First of all let us consider only the statements . The sentences are already aligned.Since A+A = A, the conclusion will be All novels are songs If we convert this conclusion, we get Some songs are novels.Hence both the conclutions given in the question are true.
eg: Statements :Some roses are leaves.
                         Some leaves are throns.
Conclusions : (i) Some roses are thorns.
                       (II) Some leaves are roses.
We know that for a combination of I+I - type no conclusion could be drawn. But if we convert the first statement, we get Some leaves are roses. Which is conclusion (ii) Also on converting the second statement, we get some thorns are leaves. This proposition is not given in the conclusion part. So in this example, conclusion (ii) alone is true.
So while solving the problems on syllogism, we should also take the immediate inferences of the given statements as well as the immediate inference of the conclusion drawn from the table.

COMPLEMENTARY PAIR
Consider the following. Conclusions :i) Some buses are trucks.
                                                             ii) Some buses are not trucks.
We know that either some buses will be trucks or some buses will not be trucks. Hence either (i) or (ii) is true. Such pair of statements are called complementary pairs. So in a complementary pair, at least one of the two statements is always true. We can call a pair as a complementary pair if
i) The subject and predicate of both the sentences are the same.
ii) They are an I + O - type pair or an A + O type pair or an I + E - type pair.
Some complementary pairs are given below.
i) All birds are swans .
  Some birds are not swans.
ii) Some tables are watches.
   Some tables are not watches.
iii) Some girls are cute.
   No girls are cute.
Note :The steps to be followed to do a syllogism problem by analytical method are mentioned below.
i) Align the sentences properly
ii) Draw conclusion using the table
iii) Check for immediate inferences.
iv) Check for complementary pair if steps ii and iii fail.

SOLVED EXAMPLES
1. Statement : No rooms are stones
                      Some houses are rooms.
Conclusions : i) Some houses are stones
                      ii) Some houses are not stones.
We can easily align the statements by changing the order of the sentences. The aligned pair is :
Some houses are rooms.
No rooms are stones.
I + E = O. So the conclusion is Some houses are not stones, Hence we obtain a definite conclusion that conclusion (ii) is correct. Hence step IV becomes unnecessary.
2. Statements :Some cows are horses
                       All cows are tigers.
Conclusions : i) Some tigers are horses.
                       ii) Some tigers are cows.
To align the sentences, it is sufficient to convert the first statement. So the aligned pair is
Some horses are cows.
All cows are tigers.
I + A = I. Hence the conclusion will be Some horses are tigers. If we convert this conclusion, we get Some tigers are horses which is conclusion (i). Also if we convert the second statement, conclusion (ii) is obtained. Hence both the conclusions given above should be taken as true. There is no need to
check for complementary pair because definite conclusion has already been obtained.
3. Statements :Some poets are teachers.
                       Some teachers are saints
Conclusions : i) Some poets are saints.
                      ii) Some poets are not saints.
This pair is already aligned. But there is no definite conclusion for I + I type combinations. Also none of the given conclusions is the immediate inference of any of the statements. So let us check for the complementary pair. The conclusions given are in the form of 'some' and 'some not'. Hence either conclusion (i) or (ii) follows.
                                  THREE - STATEMENT SYLLOGISM
This type of syllogism problems consist of 3 statements which are followed by 4 or more conclusions. A typical three - statement syllogism problem is given below.
Directions : Below are given three statements followed by several conclusions based on them.
Examine the conclusions and decide whether they logically follow from the given statements.
You have to take the given statements as true even if they appear to be at variance with
commonly known facts.
Statements : A) All bags are hats.
                    B) Some pins are bags.
                    C) No hats are needles.
Conclusions : I) Some pins are hats.
                     II) No needles are bags.
                    III) Some pins are needles.
                   IV) Some pins are not needles.
(a) Only I and II follow
(b) Only I and IV follow
(c) I, II and IV follow
(d) Either III or IV, and I follow
(e) Either III or IV and I and II follow.
Before solving this example, let us see the steps in solving a three-statement syllogism problems.
Step I
i) Consider a given conclusion.
ii) Note the subject and predicate of this given conclusion.
iii) Now find which of the two given statements has this subject and predicate.
iv) a) If there is a common term between the two statements chosen in the previous part, then consider only these two statements.
b) If there is no common term between the two statements chosen in the previous part, then we should consider all the three statements.

Step II
i) If two statements are relevant for a given conclusion, align them.
ii) If three statements are relevant, write them as a chain. That is, align them in such a way that the predicate of the first sentence and subject of the second are the same, and the predicate of the second sentence and the subject of the third sentence are the same.
iii) Now arrive at the conclusion using the table.
iv) Now compare the given conclusion with the conclusion drawn using the tables. If they match, the given conclusion is true. If they do not match, it is false.

Step III
i) If a given statement has already been marked as a valid conclusion after step II, then leave it. Otherwise check if it is an immediate inference of any of the three given statements of the conclusion
derived.
ii) Search for complementary pair :
a) Check if any two given conclusions have the same subject and the same predicate.
b) If (a) is satisfied, then check whether any of them has been marked as a valid conclusion after step II or as an immediate inference.
c) If none of them has been marked as a valid conclusion, then they will form a complementary pair if they are an A - O or I - O or I - E pair.
d) If they do make a complementary pair, then mark the choice "either of the two follows".
If a conclusion is marked as a valid conclusion after step II, then it is not necessary to perform step III (i). Again if a given conclusion has already been accepted in step III (i), then it is not necessary to perform step III
(ii). The learner should understand these steps clearly. Now follow the solution to the example which is already given. Here we have to check the validity of each and every conclusions one by one.
Conclusion I : Here the subject is pin and the predicate is hat. So let us consider (A) and (B) as our relevant statements because they have a common term 'bags'. The second step is to align the sentences.
The aligned pair is,
Some pins are bags.
All bags are hats.
I + A = I. So we arrive at the conclusion, 'Some pins are hats'. So conclusion I is valid.
Conclusion II : Here the subject is 'needles' and the predicate is 'bags'. Statement C contains the subject 'Needles'. But 'bags' appears in both A and B. We should select A because there is a common term between A and C. This is an aligned pair and so we arrive at the conclusion No bags are needles which implies No needles are bags. Hence conclusion II is valid.

Conclusion III : Here the subject is 'pins' and the 'predicate' is needles. These words appear in statements (B) and (C) respectively which have no term in common. So all the three statements
should be taken as relevant. Now align the statements as Step II (ii) So we get,
Some pins are bags
All bags are hats.
No hats are needles.
I + A + E = (I + A) + E= I + E = O.
So the conclusion is 'Some pins are not needles', which is conclusion IV. So conclusion IV is valid.
Since conclusion III is not valid in step II, let us perform step III (i). The conclusion, Some pins are not needles is not an immediate inference of any of the three given statements. So the next step is to check the existence of a complementary pair in the given conclusions.We see that conclusion III and conclusion IV form a complementary pair of I - O type. So the choice "either III or IV follows" could be selected. But we find that conclusion IV is valid from the previous step. So conclusion III is not valid. Hence for this given example, the third choice which is 'I, II and IV follow' is true.