Inequality

What is the inequality (Coded Inequality)  in math?

inequality, In mathematics, a statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions.

In these types of questions, the relations amongst the numbers are given in a coded form with a set of instructions in which different mathematical symbols replace different symbols of inequality. The examinee is expected to use the given instructions to decode the symbols and then solve the question. Let’s look at some instructions which are provided.

1. Not greater than means ‘ smaller than or equal to.’ which is ≤
2. Not smaller than means ‘ greater than or equal to.’ which is ≥
3. Neither greater than nor equal to means ‘smaller than.’ which is <
4. Neither smaller than nor equal to means ‘ greater than.’ which is >
5. Neither greater than nor smaller than means ‘equal to. Which is =

Tips and Tricks:

To solve the tricky question of inequality, the candidate must understand the four tricky concepts: 

    If in a question, K < M < L is given, then K < M, M < L and K < L are considered to be true.

    If in a question, K > M ≥ L is given, then K > L is considered to be true and K ≥ L is not true.

    If in a question, K ≥ L = M is given, in that case, either K > M or K = M is true.

    If in a question, K < M > L is given, then no relation can be found between K and L because of opposite symbols.

Types of Inequality:

a) Single statement Inequality: In this type of question, the relation between the elements is given in a single series by coded relationship symbols i.e. <, >, =, ≤, ≥ and ≠. For example;

Q. Statement: A < N = U > F > B > H

Conclusion:  I. H < N (true)            II. F > A (false)

b) Multiple statements Inequality: In this type of question, the relation between the elements is given in two or more different series. To get the exact relation, we have to arrange it by matching the similar elements in a single series. For example;

Q. Statement:  T < D > G, P < F = T

Conclusion:   I. P < G   II. G > T

Solution: Here, first, we have to arrange it in a single series to get the definite relation.

P < F = T < D > G

I. P < G (false)

II. G > T (false)

c) Not equal types Inequality: In this type of question, the ‘≠’ (not equal) relation are given between the elements. The not equal symbol is meant to show a comparison between the two quantities which are unequal hence, among the two quantities one will be either greater or smaller than the other quantity. To get the exact relation, we have to consider the both possibilities i.e. either ‘>’ or ‘<‘. For example;

Q. Statement:  T < S < D = Q, T ≠ P = X < Z < R,

Conclusion:  I. X < D (False) (≠ means either > or <)            II. Q > P (False) (≠ means either > or <)

d) Filler Inequality: In this type of question, the relation between the elements is not given and in the place of coded symbols (which represented by <, >, =, ≤, ≥ and ≠) blank or space was/were given. You have to find out the proper coded symbol/s to fill the blank or space according to a certain conditions which generally mentioned with the questions. For example;

Q. Which of the following order of letters (from left to right) in the blanks makes the expression, C > P definitely true?

____ = ____ > ____ ≤ ____=____ 

a) Z, T, C, D, P     b) T, P, D, C, Z     c) P, T, C, D, Z     d) D, C, T, Z, P     e) None 

Solution: (e) None

e) Conditional Inequality: It is an inequality which is true for some variables or for a particular condition but not true for all values of variables.  And the solution of inequality consists of only real numbers as the term “Less than or Greater than” are not defined for establish a certain relation. For example;

Q1. Which of the following statements prove that ‘W > F’ is definitely true?

I. A ≥ J < K = W ≥ L > Z; J > N = S ≤ F

II. W < P < Q ≤ K > J > Z; Z = H ≥ S = A < F;

III. S < J < W = O ≤ L < Q; S = P ≤ K < F =J

Solution: III

f) Coded Inequality: In case of coded inequality, questions consists of a couple of statements with some logical and arithmetic relationship between them.  Such type of Inequality followed by a couple of conclusions and you’ll have to find out which conclusion follows the given statements. For example;

Q. In these questions, the relationship between different elements is shown in the statement. The statement is followed by two conclusions. Choose the correct answer on the basis of the information given below.

a) If only conclusion I is true.

b) If only conclusion II is true.

c) If either conclusion I or II is true.

d) If both conclusions I and II are true.

e) If neither conclusion I nor II is true.

In the following questions, the symbols %, @, #, &, $ are used. All the symbols define the following meanings.

Y # Z means that ‘Y is equal to Z’

Y & Z means that ‘Y is greater than Z’

Y $ Z means that ‘Y is greater than or equal to Z’

Y % Z means that ‘Y is smaller than Z’

Y @ Z means that ‘Y is smaller than or equal to Z’

For example;

Q. Statements:  O $ M; P # M; R % P;

Conclusions: I) P % O  II) P # O 

Ans: c

TYPE 1 : SIGN LANGUAGE

PRACTICE TEST-I

Directions (1-4): Read the following information carefully and answer the following questions given below.

1. Statements: R=P=J>S, J=M<N

Conclusion:

1. S < R=P                           2. N>S                                 3.R=M

(a) Only 1 is true              (b) Only 2 is true              (c) Either 2 or 3 is true

(d) Both 1 and 2 are true                                             (e) All are true

2. Statements: S<P=Z=M; Z=K=R, S=J

Conclusion:       

1. M=R                                2. Z>J                                   3. S<Z=R

(a) Only 1 is true                             (b) Only 2 is true              (c) Only 1 and 2 are true

(d) All 1,2 and 3 are true               (e) Only 2 and 3 are true

3. Statements: K=M<R>P, R=S=N>J

Conclusion: 1. K<R>J       2. P>S                   3. S=K

(a) Only 1 is true                             (b) Only 2 is true                             (c) Only 1 and 3 are true

(d) All are true                                 (e) None of these

4. Statements: R=J>M, J=L=S<N, S=P

Conclusion:        1. P=J                   2. R=L                   3.M<N

(a) Only 1 is true              (b) Only 2 is true              (c) Only 1 and 3 are true

(d) All are true                  (e) None of these

Directions (5-7): Read the following information carefully. And answer the following questions given below.

5. Statements: O > V,  U ≥ T, U > 0

Conclusions:      I. U ≠ T                 II. T > U                               III. 0 ≠ T

(a) Either II or Ill are correct.        (b) Only conclusion NI is incorrect.           (c) None is correct.

(d) All conclusions are correct.    (e) Only conclusion III is correct.

6. Statements: O ≤ I > D = K; C = 0 > T; M < T

Conclusions:      I. I = M                 II. I > M

(a) None is true                (b) Both I and II are true               (c) Only II is true

(d) Only I is true                              (e) Either I or II is true

7. Statements: T ≤ Z < I, G = J > T, F = O < G

Conclusions:      I. G > Z                 II. Z ≥ G

(a) If only conclusion I is true.     (b) If only conclusion 11 is true. (c) If either 1 or II is true.

(d) If neither I nor II is true.          (e) If both 1 and II are true.

Directions (8): In the following question assuming the given statements to be true, find which of the conclusion among the given conclusions definitely true and then give your answers accordingly.

8. Statements: l ≥ Q > 2; Q > S > T; 2 > U; V < U

Conclusion:        I. U < I                  II. 2 < T                 III. 1 > T               IV. 1 > U  

(a) Only III is True                                                         (b) Only conclusions I & 11 are True        

(c) Only conclusions 1, 111 and IV are True           (d) Either conclusion II or III is True         

(e) All conclusions are True

Directions (9): In the following question assuming the given statements to be True, find which of the conclusion among given conclusions is / are definitely true and then give your answers accordingly.

9. Statements: @ ≤ 4 ≥ C ≥ N, C > X ≤ 1, @ = R, X ≤ 2 > T

Conclusions:      I.4 < l     II.T < X                  III. X < 4

(a) None is True               (b) Both 1 and II are True             (c) Only III is True

(d) Only I is True               (e) None of these

Direction (10): In the following question assuming the given statements to be True, find which of the conclusion among given conclusion(s) is/are definitely true and then give your answers accordingly.

10. Statements: 1 ≥ 4; B < C < E; F < C; 4 = B

Conclusions:      I. F < E                  II. 1 > C                III. F > 4                               IV.B > l

(a) Only II is True                             (b) Only I is True                              (c) Only I and III are True

(d) Only II and IV are True            (e) Only IV is True

1. (e)     2. (d)     3. (a)      4. (d)     5. (c)      6. (c)      7. (c)      8. (c)      9. (c)      10. (b)

TYPE II: DERIVING THE APPROPRIATE CONCLUSION

In this type of questions, certain relationships between different sets of elements is given, using either the real symbols or substituted symbols. The candidate is required to analyse the given statements and then decide which of the relations given as alternatives follows from those given in the statements.

Rules helpful in solving such problems

Rule 1. First see, if the two inequalities have a common term. Go to next step only if they have the common term (otherwise don’t).

Rule 2. If the common term is greater than or equal to (?) one term, and less than or equal to (`<‘) other one, i.e., if it is greater than or equal to both (or less than or equal to both), a combination is not possible.

Rule 3. Combine the two inequalities and draw a conclusion by letting the middle term disappear. The conclusion will normally have a ‘>’ (or a 1<‘) sign strictly, unless the `>: sign (or appears twice in the combined inequality.

Rule 4. The relationship represented by sign or `<‘ can only be established between two terms, if the common term is preceded as well as succeeded by the same sign.

Rule 5. If the common term is preceded by and followed by > i.e., A> B > C, then the relation between A and C call only be : A > C. because common term is only proceeded by ‘>’ and is not followed by the same sign again.

The solution requires that we should follow the following steps

Step I. From the given equation. first of all. take one symbol or coded relation and change the same with the inequality sign in all questions.

Step 2. In the given term, look for the common term and keeping it in the middle, join the other two terms with it. Eg. If we are given two terms A > B and B > C, we know that B is common term and keeping it in the middle we can make one single equation as A > B > C. We may note that the common term is followed as well as succeeded in the same sign.

Step 3. From the statement A > B > C, we can conclude that A > C, because, B is preceded by and succeeded by the sign of > (Rule 2).

Important Points to Remember

I. Alter decoding and locating the common term, take one conclusion at a lime and find out which statements are relevant to find out the conclusion. The term appearing in the ‘conclusion to be verified’ appear along with the common term in the relevant statements.

2. Check whether the conclusion follows from only one given statement. For example, we may be given that P is greater than Q (P > Q). If we are given the conclusion Q is less than P (Q < P), we need not look for solution in any other statement because both these statements are identical and follow from each other.

3. If in our conclusion, we arrive at conclusion A > B, then it automatically follows that B < A. For example, if we get the conclusion that A ≥ C then the conclusion C ≤ A obviously follows.

4. If our conclusion after third step is A ≥ B and the given conclusion to be verified are (i) A= B (ii) A> B, then the choice is “either i or ii follow”

5. The meaning of the terms:

• neither equal to nor greater than’ is ‘<‘ i.e., less than

• neither equal to nor less than is ‘>’ i.e., greater than

• Equal to or greater than is ‘>’                                 • Equal to or less than is `<‘

• Not less than means ‘>.                                            • Not greater than means ‘<‘

• Neither greater than or less than means

6. If from the given terms in the statement, no relation can be established between the terms mentioned in the conclusion, but the given two conclusions are absolutely contradictory, then either of the two conclusion is correct and our answer should be ‘either I or II follows’.

SOLVED EXAMPLES

Ex.1. Directions (Ex. 1 to 3): In the following questions the symbols @, ©, $, % and * are used with the following meaning as illustrated below:

P © Q means P is not greater than Q.

P % Q means P is not smaller than Q.

P * Q means P is neither smaller than nor equal to Q.

P @ Q means P is neither greater than nor equal to Q.

P $ Q means P is neither greater than nor smaller than Q.

Now in each of the following questions, assuming the given statements to be true, find which of the conclusions I and 11 given are/is defiantly true.

Give answer (a) if only conclusion I is true.

Give answer (b) if only conclusion II is true.

Give answer (c) if either conclusion I or II is true.

Give answer (d) if neither conclusion I nor II is true.

Give answer (e) if both conclusion I and II are true.

1. Statements: K@V. V©N, N%F

Conclusion:        I. F @ V                II. K @ N

Sol. (b) :              (i) K < V ;              (ii) V ≤ N              (iii) N ≥ F.

from (iii) and (ii), F and V can’t be compared, hence I cannot be followed.

From (i) and (ii), K < V < N or K < N. Hence II follows

2. Statements: H © W, W$M, M@B

Conclusion:        I. B*H    II. M%H

Sol. (e) :  (i) H ≤ W (ii) W = M, (iii) M < B

Combining these, we get H ≤ W = M<B Hence B > H and I follows. Also, M ≥ H and II follows.

3. Statements: R$J, J%D, D*F

Conclusions: 1. D$R 11. D@R

Sol. (c): (i) R = J, (ii) J ≥ D, (iii) D > F

Combining these, we get R = J ≥ D > F. Hence D ≤ R. So either I. (D = R) or II. (D < R) follows.

PRACTICE TEST-II

Directions (1 to 3) : In the following questions, the symbols are used with the following meanings as illustrated below :

P$Q means `P is smaller than Q’;

P*Q means ‘P is neither smaller than nor greater than Q’;

P#1Q means `P is either greater than or equal to Q’;

P%Q means ‘ P is greater than Q’;

P©Q means ‘P is either smaller than or equal to Q’;

Now, in each of the following questions, assuming the given statements to be true, find which of the two conclusions I and II given below then is/are definitely true?

Give answer

(a) if only conclusion I is true;                    (b) if only conclusion II is true;

(c) if either conclusion I or II is true;         (d) if neither conclusion I nor II is true;

and (e) if both conclusion I and II are true.

1. Statements: B#D, D*F, F%H

Conclusions: I. F*B                         II. F$B

2. Statements M%K, K#T, T*J

Conclusions: I. J@K                        II. T$M

3. Statements: V©R, R$M, M*W

Conclusions: 1. W%V                     II. V©W

Directions (4-8): In these questions, relationship between different elements is shown in the statements. These statements are followed by five conclusions. Read the statements and then decide which of the following conclusion follow from the given statements.

4. In w hick of the following expression will the expression ‘A < B’ does not hold true?

(a) A≤ G = N ≤ B ≤ H                        (b) G > J = B ≥ O > N = A                (c) N < B = H ≥ J = A

(d) B < G ? H = L ? A = P                 (e) A ≤ H = G ≤ B < J

5. In which of the following expressions will the expression ‘N > J’ Definitely be true?

(a) N > M ≤ K < J ≤ L                        (b) J ≤ K = M < N ≤ L                        (c) J ≤ M = K < L ≤ N                       

(d) L ≤ J < M > N = K                        (c) Both 2 and 3

6. Which of the following expressions will be true if the given expression `A> Y > X <W <V’ is definitely true?

(a) A > W             (b) V ≥ X              (c) Y ≥ W              (d) A > X              (e) Y > V

7. Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the given expression in such a manner that “7 < T” definitely holds True?

R __ 7 __ Y __ 0 __ T

(a) =, ≤,  ? , >                     (b) >,  <, = , >                     (c) <, =,  ≤, <        (d) ≤, ≤, =, <        (e) None of these

8. In w hick of the following expressions will the expression ‘6 > S’ as well as ‘R < U’ be definitely true?

(a) 6 = Q > R < S < T ≤ U                (b) 6 ≥ Q = R > S > T < U                (c) 6 > Q < R = S < T = U

(d) 6 ≥ Q ≥ R = S < T < U                (e) 6 > Q ≥ R = S < T ≤ U

Directions (9 to 13): In earls of the following questions symbols $, %, * and @ are used with following meanings:

X$Y means X is not greater than Y;

X#Y means X is neither greater than nor smaller than Y;

X%Y means X is not smaller than Y;

X*Y means X is neither smaller than nor equal to Y;

X@Y means X is neither greater than nor equal to Y;

Now, in each of the following questions, assuming the given statements to be true, find which of the three conclusions I, II and III given below them is/are definitely true.

9. Statements: D$K, H*B. K@H

Conclusions: I. B%K.        II. B @ K               III.  H * D

(a) Only I and II are true                               (b) Only either I and II are true  (c) Only I and Ill are true

(d) Only either I, II or III are true                 (e) None of these.

10. Statements: R$G, G*B

Conclusions: I. T@ B       II.  B * R III. T $ G

(a) None is true (b) either I or II is true     (C) Only I and III are true

(d) Only either I, II or III are true                (e) Only I and either II or 111 are true.

11. Statements: F#M, M*J, P % F

Conclusions: 1. P*J          II. P % J I              ll. P#M

(a) Only I is true                              (b) Only I and II are true                (c) Only I and III are true

(d) Only II and III or I are true      (e) None of these.

12. Statements: L % J, . L@ K, J*F

Conclusions:      I. F@K                  II. K*J    Ill. F @ L.

(a) None is true (b) Only I and II are true                (c) Only II and III are true

(d) Only I and Ill are true (e) All are true.

13. Statements: N$P, P @ Q, H%Q

Conclusions: I. H%N        II. N@H                III. N#H

(a) Only I is true   (b) Only II is true           (c) Only I and II are true  (d) All are true   (e) None of these.

ANSWERS

 1.(c)      2. (e)     3. (a)      4. (d)     5. (e)     6. (d)     7. (c)       8. (e)    9. (e)     10. (a)  11. (a)     12. (e)          13. (b)

PRACTICE TEST-Ill

Directions (1 to 5): In the following questions, the symbols are used with the following meanings as illustrated bellow:

P%Q means P is not smaller than Q.

P$Q means P is not greater than Q.

P#Q means P is neither greater than nor equal to Q.

P©Q means P is neither smaller than nor equal to Q.

P@Q means P is neither greater than nor smaller than Q.

1. Statement: J©M, M#K, K % D

Conclusions: I. J © D                      II. D#M                 III. K © J

(a) None is true  (b) Only I is true (c) Only Il is true (d) Only III is true (e) Only I and II are true.

2. Statements: R@K, K$F, F#N

Conclusions: I. N©R        II. F @ R Ill. F@R

(a) Only I is true (b) Only either II or III is true (d) Only III is true (e) None of these.

3. Statements: R$D, D@N. N©F

Conclusions: I. F#D          11. F@R                              III. N%R

(a) Only I and II are true (b) Only I and III are true              (c) Only II and III are true

(d) All are true (e) None of these.

4. Statements: H # T, T $ B, B  © R

Conclusions: I. R © H      II. B © H              Ill. T # R

(a) Only I is true (b) Only I and II are true (c) Only I and III are true

(d) Only II and III are true             (e) None of these.

5. Statement:  M % D, D # K, K  $ N

Conclusions: I. K © M                    II. N @ D                             III. M © N

(a) Only I is true               (a) Only II is true     (d) Only III is true (d) Only I and II are true

(e) None follows.

Directions (6 to 10) : In these questions symbols are used with the following meanings:

P@Q means P is not smaller than Q.

P#Q means P is not greater than Q.

P%Q means P is neither greater than nor equal to Q.

P$Q means P is neither smaller than nor equal to Q.

P©Q  means P is neither greater than nor smaller than Q.

Now, in each of the following questions assuming the given statements to be true, find which of the following three conclusions I, II and III is/are definitely true and give your answer accordingly.

6. Statements: M % R,  R#T, T©N

Conclusions: I. N©R                       II. N$R                  III. N$M

(a) All follow       (b) Only either I or II follows        (c) Only either I, II or III follows

(d) Only either I or III and II follows                         (e) None of these.

7. Statements:  J # N, K@N, T$K

Conclusions:      I. J%T                    II. T$N                  Ill. N @ J

(a) None follows (b) Only l and II follows (c) Only I and Ill follows (d) Only II and Ill follows (e) All follows.

8. Statements: B@D, D @ H, H % F

Conclusions: I. B@F                       II. D@H                III. D$F

(a) None follows (b) either l or II follows (c) Only either I or II or Ill follows (d) Only II and Ill follows (e) None of these

9. Statements:  T$K. K#R. R © M

Conclusion: I. M© K                       ll. M % T              III. M $ K

(a) All follows (b) Only either I or Ill follows (c) Only either I or II follows (d) Only either II or III follows

(e) None of these.

10. Statements:  V@M, A$M. R#V

Conclusions:      I. R#A                   II. V@A                 III. R$M

 (a) Only I follow (b) Only II follows (c) Only Ill follows       (d) None of follows         (e) All follows

Directions (11-12): In a certain coding system,

P & Q means P is not greater than Q.

p % Q means P is neither smaller than nor greater than Q.

P @ Q means P is neither smaller nor equal to Q.

P $ Q means P is not smaller than Q.

Based on the information provided above, answer the following. questions.

11. Statements: l @ 3 & D, Q % R @ l, S % 3

Conclusions:      I. 3 % Q                II. D @ S

(a) Only I follows              (b) Only Il follows            (c) Either I or II follows (d) Neither I nor II follows

(e) Both I and II follow

12. Statements: 1 $ V, 1 % G, M @ G

Conclusions: I. M @ D                   II. D  % M

(a) Only I follows              (b) Only Il follows            (c) Either I or II follows (d) Neither I nor II follows

(e) Both I and II follow

Directions (13-15): In the following questions, the symbols $, %, *, & and © are used with the following meaning as illustrated below:

‘Z % B’ means ‘Z is greater than B’.

‘Z $ B’ means ‘Z is not greater than B’.

‘Z * B’ means ‘Z is neither greater than nor equal to B’.

‘Z & B’ means ‘Z is either greater than or equal to B’.

‘Z© B’ means ‘Z is neither smaller than nor greater than B’.

Now in each of the following questions assuming the given statements to be true, find which of the conclusion/s given below them is/are definitely true?

13. Statements: 2* 1, l $ M, P & M, M $ 3

Conclusions: I. P% 2                       II. L $ P                               III. P % 3

(a) Only Conclusion I is true. (b) Both Conclusions I and II are true. (c) Either Conclusion I or Ill is true.

(d) Neither Conclusion I nor III is true.     (e) Either Conclusions I or II and III are true

14. Statements: A $ S © D, Z & X © A, C % S

Conclusions: I. S % Z                      II. D * C                III.X * C

(a) Only Conclusion I is true. (b) Both Conclusions I and II are true. (c) Either Conclusion I or Ill is true.

(d) Neither Conclusion I nor III is true.     (e) Either Conclusions I or II and III are true

15. Statements: U©G, U % S & V, B $ S

Conclusions: I. U % V                     II. G © S                              III. V % B

(a) Only Conclusion I is true. (b) Both Conclusions I and II are true. (c) Either Conclusion I or Ill is true.

(d) Neither Conclusion I nor III is true.     (e) Either Conclusions I or II and III are true

ANSWERS

I. (a) 2. (c) 3. (b) 4. (e) 5. (e) 6. (c)  7. (e) 8. (a) 9. (b) 10. (d) 11. (d) 12. (d) 13. (b) 14. (b) 15 (a)

PRACTICE TEST-IV

Directions (1 to 5): In the following questions, the symbols are used with the following meaning

A $ B means ‘A is not smaller than B’

A # B means ‘A is neither greater than nor equal to B’

A @ B means ‘A is neither smaller than nor equals to B’

A % B means ‘A is not greater than B’

A * B means ‘A is neither greater than not smaller than B’.

In each of the following questions assuming the given statement to be true, find out which of the three conclusions I, II and III given below is/are definitely true.

1. Statement:  D * Q, Q @ L, L $ B, B # G.

Conclusion: I. D @ B                      II. B * D                III. G @ L

(a) Either I of II only         (b) I and II only                 (c) I only              (b) II and III only  (e) None is true.

2. Statement: Z @ y, y # K, K % M, M @ T

Conclusion: I. Z @ M                     II. Y @ T                              III. Z # K

(a) I only (b) II and III only   (c)     III only (c) III only (e) None is true.

3. Statement: P # M, M % R, R * T, T # L

Conclusion:        I. P # R                  II. P * R III. M % L

(a) I only              (b) Either (I) or (II)          (c) (III) only         (d) All are true   (e) None is true.

4. Statement: F @ H, M % H, M $ R, G * M

Conclusion: I. F $ R                         II. F @ R                              III. H $ G

(a) (II) and (III) only          (b) (II) only          (d) (III) and either (I) or (II)           (e) None is these.

5. Statement: K @ T, T # D, D * F, F % G

Conclusion:        I. G @ R II. G * T                              III. G @ T

(a) I and II only (b) II and III Only              (b) Either II or III Only      (d) Ill only            (e) None is true.

Directions (6 to 15): In the following questions, the symbols are used with following meaning.

P © Q means P is neither greater than nor smaller than Q’.

P # Q means ‘P is either greater than or equal to Q’.

P $ Q means P is either smaller than or equal to Q’

P % Q means ’13 is greater than Q’

P @ Q means ‘P is smaller than Q’.

In each of the following questions assuming the given statements to be true, find out which of the two conclusions I and II given below them is/are definitely true.

(a) If only conclusion I is true (b) If only conclusion II is true (c) If either conclusion I or II is true

(d) If neither I or II is true (e) If both I and II are true

6. Statement: M $ J, J @ R, R © D             Conclusion: I. D % M       II. R % M

7. Statement: N # T, T © F, F % B               Conclusion: I. F $ N          II. B $ T

8. Statement: H @ J, M % J, M # K             Conclusion: I. K % J          II. M % H

9. Statement: V $ T, T # R, R @ B               Conclusion: I. V @ B        II. B % T

10. Statement: R % M, M # B, B $ D          Conclusion: I. D % R        II. D $ R

11. Statement: E © F, F @ K, K @ J           Conclusion: I. J % F          II. K % E

12. Statement: B # N, N % D, D $ W          Conclusion: I. W % B       II. N % W

13. Statement: T @ B, B $ K, K % M           Conclusion: I. K % T        II. M % B

14. Statement: H © T, T @ M, M # J         Conclusion: 1. J @ H       II. J # H

15. Statement: R $ M, N © M, D # N        Conclusion: 1. D ©R        II. D # M

Direction (16 – 20): Read the following information carefully. And answer the following questions given below.

‘6 % 10’ means ‘6 is not greater than 10’.

‘6 δ 10′ means ‘6 is neither greater than nor smaller than 10’.

‘6 # 10’ means ‘6 is neither greater than nor equal to 10’.

‘6 © 10’ means ‘6 is not smaller than 10’.

‘6 @ 10’ means ‘6 is neither smaller than nor equal to 10’.

Now, in each of the following questions assuming the given statements to be true, find which of the two Conclusions I and II given below them is/are definitely true?

16. Statements: L © G, G @ N, N δ X

Conclusions:      I. X # G                                II. N # G

 (a) If only Conclusion 1 is true    (b) If only Conclusion II is true     (c) If either Conclusion I or II is true

(d) If neither Conclusion I nor II is true (e) If both Conclusion I and II are true

17. Statements: L @ M, M % I, I © V

Conclusions:      I. L @ V                               II. M # V

(a) If only Conclusion I is true      (b) If only Conclusion II is true     (c) if either Conclusion I or II is true

(d) If neither Conclusion I nor II is true    (e) If both Conclusion I and II are true

18. Statements: W # G, G @ I, I © T

Conclusions: I. T # W                                                   II. T # G

(a) If only Conclusion I is true      (b) If only Conclusion II is true     (c) If either Conclusion I or II is true

(d) If neither Conclusion I nor II is true     (e) If both Conclusion I and II are true

19. Statements: N δ S, S % W, W # I

Conclusions: I. W © N                                  II. I © S

(a) If only Conclusion I is true      (b) If only Conclusion II is true     (c) If either Conclusion I or II is true

(d) If neither Conclusion I nor II is true                   (e) If both Conclusion I and II are true

20. Statements: D @ T, M © T, M % B

Conclusions:      I. D @ M                             II. B © T

(a) If only Conclusion I is true (c) If either Conclusion I or II is true (e) If both Conclusion 1 and II are true

(b) If only Conclusion 11 is true (d) If neither Conclusion I nor 11 is true

ANSWERS

1. (c)      2. (e)     3. (a)      4. (a)      5. (d)     6. (e)     7. (a)      8. (b)     9. (d)     10. (c)  

11.(e)    12. (d)   13. (a)   14. (c)   15. (b)   16. (e)   17. (d)   18. (b)   19. (a)   20. (b)