2. Mathematical logic
2.1 Introduction
In mathematical logic, a symbol or an operator is given and the operation which could be
performed using that symbol or operator is also defined. This could be explained using following example.
A O B = A + B
Here, symbol O is defined as +. The question that could be asked are as follows:
[1] Find the value of 5 O 6.
5 O 6 = 5 + 6 = 11
[2] Compare the given expressions.
e.g., Which is the greater: 6 O 6 or 9 O 3?
6 O 6 = 12 and 9 O 3 = 12
\ 6 O 6 = 9 O 3
Note: While solving these kind of problems one has to remember that the operation would be
performed as it is defined in the problem given. Sometimes a conventional operator with different meaning may be given.
Then one has to perform the operation that is defined and not the conventional one.
i.e., if A × B = A + B
Then, value of 3 × 4 = 3 + 4 = 7 and not 3 × 4 = 12.
Directions for Illustration 1 to 5: Refer to the data below and answer the questions that follow.
A + B means A × B.
A ÷ B means A – B.
A × B means A + B.
A – B means A ÷ B.
Illustration 1
Find the value of 28 – 4 × 3 ÷10.
Solution
28 – 4 × 3 ÷ 10 Þ 28 ÷ 4 + 3 – 10 = 7 + 3 – 10 = 0
Illustration 2
Find the value of 63 × 4 + 3 × 75 – 5.
Solution
63 × 4 + 3 × 75 – 5 Þ 63 + 4 × 3 + 75 ÷ 5 = 63 + 12 + 15 = 90
Illustration 3
Find the value of 5 × 5 + 1 × 5 + 2 × 5 + 3.
Solution
5 × 5 + 1 × 5 + 2 × 5 + 3 Þ 5 + 5 × 1 + 5 × 2 + 5 × 3 = 5 + 5 + 10 + 15 = 35
Illustration 4
Which of the following is true?
(i) a × b ÷ c = a ÷ c × b
(ii) a × b + c = a × c + b
(iii) a + b × c = a × b + c
Solution
(i) a × b ÷ c Þ a + b – c and a ÷ c × b Þ a – c + b
\ a × b ÷ c = a ÷ c × b. Hence, (i) is true,
(ii) a × b + c Þ a + b × c = a + bc
a × c + b Þ a + c × b = a + bc
\a × b + c = a × c + b. Hence, (ii) is true
(iii) a + b × c Þ a × b + c = ab + c
a × b + c Þ a + b × c = a + bc. Hence, (iii) is not true.
Illustration 5
Find the value of
Solution
Þ = = = 1
Directions for illustration 6 to 9: Refer to the data below and mark the alternative that
follows from the data given in the question.
‘=’ means ‘<’ , ‘D’ means ‘>’ , ‘£’ means ‘£’ , ‘’ means ‘³’ , means ‘=’
Illustration 6
a + b c + d a + h.
(A) a = h (B) b ³ c (C) b ³ h (D) c + d = a + b (E) None of these
Solution
a + b ³ c + d = a + h Þ a + b ³ a + h \ b ³ h. Hence, [c].
Illustration 7
a D b = c d
(A) c > a (B) b < d (C) c = a (D) b = d (E) None of these
Solution
a > b < c = d a may or may not be greater than c. b < d. Hence, [b].
Illustration 8
a2 = b2 c d2
(A) a2 < d2 (B) b < a (C) b2 ³ d2 (D) c2 = d2 (E) None of these
Solution
a2 < b2 ³ c = d2 ; b2 ³ d2. Hence, [c].
Illustration 9
0 = a + x – z2 = 81.
(A) a < z (B) a2 < x2 (C) z < a (D) x > z2 (E) None of these
Solution
0 < a + x – z2 < 81 ; z2 < a + x
Nothing can be said about z2 and a or x as a, x and z may be +ve or -ve. Hence, [E].
EXERCISE-I
Directions: In questions (1 to 8) the following symbols have been used
× stands for equal to < stands for not equal to – stands for greater than
+ stands for not greater than > stands for less than = stands for not less than
1. If p = q + r, then it is possible that
(A) p × q – r (B) p + q –r (C) p – q –r (D) p < q < r (E) p > q + r
2. If p > q × r, then it is possible that
(A) p + r + q (B) p = r –q (C) p × q + r (D) p = q – r (E) p – q × r
3. If p × q × r, then it is not possible that
(A) p + q = r (B) p = q + r (C) p + q + r (D) p = q = r (E) p < q = r
4. If p – q + r, then it is possible that
(A) p = q = r (B) p < q – r (C) p + q × r (D) p × q × r (E) p > q > r
5. If p + q – r, then it is not possible that
(A) p × q = r (B) p + q < r (C) p < q = r (D) p = q = r (E) p – q – r
6. If p < q < r, then it possible that
(A) p + q × r (B) p – q × r (C) p > q = r (D) p × q = r (E) p × q > r
7. If p + q = r, then it is not possible that
(A) p + q × r (B) p × q > r (C) p < q + r (D) p = q – r (E) p > q < r
8. If p = q = r, then it is possible that
(A) p × q × r (B) p < q > r (C) p > q + r (D) p > q > r (E) p + q > r
Directions: In questions (9 to 12) using the symbols given below find which of the five options
in each question describes the correct relationship?
× stands for addition < stands for substraction > stands for multiplication
+ stands for division O stands for greater than = stands for less than
– stands for equal to
9. (A) 8 < 4 × 3 – 3 × 2 × 1 (B) 8 > 4 < 3 – 3 > 2 > 1
(C) 8 + 4 × 3 = 3 > 2 × 1 (D) 8 + 4 < 3 O 3 < 2 < 1
(E) 8 > 4 × 3 – 3 > 2 × 1
10. (A) 5 > 2 < 1 – 3 × 4 × 1 (B) 5 < 2 × 1 O 3 > 4 × 1
(C) 5 × 2 < 1 –3 < 4 × 1 (D) 5 + 2 × 1 = 3 + 4 > 1
(E) 5 > 2 × 1 – 3 > 4 < 1
11. (A) 5 + 3 < 7 – 8 × 4 + 2 (B) 5 × 3 < 7 O 8 + 4 < 2
(C) 5 > 3 × 7 = 8 > 4 + 2 (D) 5 < 3 > 7 – 8 > 4 + 2
(E) 5 × 3 + 7 O 8 > 4 × 2
12. (A) 3 + 4 < 2 – 5 × 2 < 1 (B) 3 × 4 × 2 = 5 + 2 × 1
(C) 3 < 4 > 2 O 5 > 2 + 1 (D) 3 > 4 × 2 O 5 > 2 > 1
(E) 3 × 4 + 2 O 5 × 2 × 1
Directions: In questions (13 to 17)
Below is given a new set of symbols for the four fundamental operations and three relations of
>, < and =
Ú stands for addition ^ stands for subtraction ( stands for multiplication
) stands for division È stands for equal to Ç stands for greater than
O stands for less than
Find which of the five options in each question given below is correct
13. (A) 24 (3 Ú 4) 2 Ç 8 (B) 24 (3 È 4) 2 Ú 8
(C) 24 ) 3 Ú 4 O 2 ^ 8 (D) 24 ^ 3 ) 4 ^ 2 ( 8
(E) 24 ) 3 Ç 4 È 2 ( 8
14. (A) 24 ) 3 Ú 4 Ç 2 ( 8 (B) 24 ( 3 ) 4 ^ 2 O 8
(C) 24 ^ 3 È 4 ) 2 Ú 8 (D) 24 ) 3 Ç 4 ( 2 ^ 8
(E) 24 Ú 3 O 4 ( 2 ^ 8
15. (A) 24 Ç 3 ( 4 Ú 2) 8 (B) 24 È 3 Ú 4 ^ 2( 8
(C) 24 ( 3 O 4 ) 2 Ú 8 (D) 24 O 3 ( 4 ^ 2 Ú 8
(E) 24 ^ 3 Ú 4 È 2 ( 8
16. (A) 24 Ú 3 ( 4 ) 2 È 8 (B) 24 ) 3 ( 4 Ç 2 Ú 8
(C) 24 ^ 3 ) 4 È 2 Ú 8 (D) 24 ^ 3 È 4 ) 2 ( 8
(E) 24 ^ 3 )4 O 2 ( 8
17. (A) 24 (3 Ú ) 2 O 8 (B) 24 ) 3 Ç 4 Ú 2 ( 8
(C) 24 ( 3 È 4 ^ 2)8 (D) 24 Ú 3 (4 Ç 2 Ú 8
(E) 24 È 3 ( 4 ) 2 Ú 8
In questions (18 to 27) the four fundamental operations and three relations namely
>, < and = are given new symbols and are defined as follows:
> stands for division Ú stands for multiplication < stands for addition
^ stands for subtraction + stands for equal to – stands for greater than
× stands for less than
Only one of the expressions as given in the five alternatives has correct relation.
Find that alternative
18. (A) 6 ^ 2 Ú 3 > 8 < 4 – 13 (B) 6 < 2 > 3 ^ 8 Ú 4 + 13
(C) 6 > 2 Ú 3 < 8 ^ 4 + 13 (D) 6 Ú 2 < 3 > 8 ^ 4 – 13
(E) 6 Ú 2 < 3 ^ 8 > 4 × 13
19. (A) 6 Ú 3 ^ 2 > 4 < 8 × 13 (B) 6 Ú 3 > 2 < 4 ^ 8 × 13
(C) 6 < 3 > 2 ^ 4 Ú 8 – 13 (D) 6 > 3 Ú 2 < 4 ^ 8 – 13
(E) 6 ^ 3 < 2 > 4 Ú 8 + 13
20. (A) 8 ^ 4 > 6 Ú 2 < 3 – 13 (B) 8 Ú 4 ^ 6 > 2 < 3 + 13
(C) 8 < 4 Ú 6 ^ 2 > 3 × 13 (D) 8 < 4 Ú 6 > 2 ^ 3 – 13
(E) 8 > 4 < 6 ^ 2 Ú 3 – 13
21. (A) 6 > 2 ^ 4 < 8 Ú 3 + 4 (B) 6 ^ 2 > 6 Ú 8 < 3 × 4
(C) 6 Ú 2 < 4 > 8 ^ 3 + 4 (D) 6 > 2 Ú 4 ^ 8 < 3 × 4
(E) 6 < 2 Ú 4 > 8 ^ 3 + 4
22. (A) 8 < 2 > 6 Ú 3 ^ 4 – 7 (B) 8 > 2 ^ 6 < 3 Ú 4 + 7
(C) 8 ^ 2 Ú 6 > 3 < 4 – 7 (D) 8 < 2 ^ 6 > 3 Ú 4 × 7
(E) 8 ^ 2 < 6 Ú 3 > 4 × 7
23. (A) 8 ^ 4 < 2 > 6 Ú 3 – 17 (B) 8 > 4 Ú 2 ^ 6 < 3 + 17
(C) 8 ^ 4 Ú 2 < 6 > 3 – 17 (D) 8 Ú 4 > 2 ^ 6 < 3 × 17
(E) 8 < 4 > 2 Ú 6 ^ 3 × 17
24 (A) 6 > 2 ^ 8 < 12 Ú 3 × 6 (B) 6 > 2 Ú 8 ^ 12 < 3 – 6
(C) 6 < 2 ^ 8 Ú 12 > 3 – 6 (D) 6 < 2 > 8 Ú 12 ^ 3 + 6
(E) 6 Ú 2 ^ 8 > 12 < 3 × 6
25 (A) 9 Ú 3 ^ 8 < 4 > 2 + 15 (B) 9 > 3 < 8 Ú 4 ^ 2 × 15
(C) 9 ^ 3 Ú 8 > 4 < 2 × 15 (D) 9 < 3 > 8 ^ 4 Ú 2 – 15
(E) 9 < 3 ^ 8 > 4 Ú 2 – 15
26. (A) 8 < 16 ^ 4 Ú 12 > 3 + 8 (B) 8 > 16 Ú 4 ^ 12 < 3 + 8
(C) 8 > 16 < 4 ^ 12 Ú 3 – 8 (D) 8 ^ 16 > 4 < 12 Ú 3 × 8
(E) 8< 16 Ú 4 > 12 ^ 3 × 8
EXERCISE-II
2. If ‘A’ means ‘+’, ‘B’ means ‘–’, ‘C’ means ‘×’ and ‘D’ means ‘¸’, then
16 A 14 C 6 D 3 B 6 = ?
(A) 108 (B) 42 (C) 38 (D) 25 (E) None of these
3. If ‘a’ means ‘+’, ‘b’ means ‘–’, ‘C’ means ‘¸’ and ‘d’ means ‘×’, then
25 c 5 d 8 b 10 a 15 = ?
(A) 55 (B) 40 (C) 44 (D) 50 (E) None of these
4. If × means +, + means ×, ¸ means – and – means ¸, then
25 – 5 × 3 + 2 ¸ 8 = ?
(A) 3 (B) 13 (C) 7 (D) –5 (E) None of these
5. If × means –, – means ×, + means ¸ and ¸ means +, then
24 × 5 – 2 ¸ 8 + 4 = ?
(A)13 (B) 50 (C) 8 (D) 16 (E) None of these
6. If – menas ¸, ¸ means –, + means × and × means +, then
12 – 4 × 7 + 8 ¸ 5 = ?
(A) 51 (B) 45 (C) 34 (D) 4 (E) None of these
7. If + means, ×, × means +, – means ¸ and ¸ means –, then
16 × 2 ¸ 4 + 7 – 8 = ?
(A) 31 (B) 29/2 (C) 43/2 (D) 15 (E) None of these
8. If + means ¸, ¸ means –, – means × and × means +, then
64 + 8 ¸ 6 – 4 × 2 = ?
(A) 34 (B) 16 (C) –14 (D) 24 (E) None of these
9. If + means –, – means ×, × means ¸ and ¸ means +, then
48 × 4 ¸ 7 + 8 – 2 = ?
(A) 3 (B) –5 (C) 35 (D) 16 (E) None of these
10. If + means –, – means ×, × means ¸ and ¸ means +, then
16 ¸ 4 × 2 – 5 + 8 = ?
(A) 58 (B) 50 (C) 44 (D) 42 (E) None of these
DIRECTIONS for questions 11 to 15: Refer to the data below and
answer the questions that follow.
A D B = ; A O B = ; A £ B = ; A * B =
11. Find the value of (2 D 8) O .(6 D 4).
(A) 10 (B) 20 (C) 0 (D) 2 (E) None of these
12. Find the value of. 12 £ 6.
(A) 5 (B) 12 (C) 18 (D) 3 (E) None of these
13. Find the value of (10 £ 6) D (28 £ 20).
(A) 5 (B) 6 (C) 8 (D) 10 (E) None of these
14. Which of the options is greatest?
(A) (2 D 3) O (2 D 3) (B) (7 £ 3) D 4
(C) (7 * 4) D (7 £ 4) (D) (E) (4 D 5) * (5 O 4)
15. Which of the following is true?
i. A D B = – (A O B) ii. A £ B = iii. A * B =
(A) i and ii (B)ii and iii (C) i and iii (D) i, ii and iii (E) None of these
Directions for questions 16 to 20: Refer to the data below and mark the alternative
thai follows from the data given in the question.
‘=’ means ‘<‘ ; ‘D’ means ‘>’; ‘£ means ‘£’ , ‘’ means ‘³’ ; ‘’ means ‘=’
16. x2 + b2 D z2 + a2; b a
(A) x > z (B) b2 = a2 (C) x2 = z2 (D) x + b = z + a (E) None of these
17. ab = c
(A) a > z (B) ab < c (C) (ab)2 < z (D) c < ab (E) None of these
18. AB D (Z + R) £ M.
(A) AB £ M (B) Y £ MN (C) Z + R>AB (D) – R > Z (E) None of these
19. £ Z2 1.
(A) Z = 1 (B) PQ £ Z (C) PQ £ ± R (D) PQ ³ Z2R (E) None of these
20. £ L = Z2 M.
(A) M + N > Z2 K (B) L < M (C) KL > Z2 (D) ³ L – N (E) None of these
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Answers
Exercise-I
1. (D) 2. (A) 3. (E) 4. (A) 5. (E)
6. (C) 7. (B) 8. (A) 9. (C) 10. (E)
11. (B) 12. (D) 13. (A) 14. (D) 15. (A)
16. (B) 17. (D) 18. (C) 19. (B) 20. (D)
21. (E) 22. (C) 23. (D) 24. (B) 25. (C)
26. (A) 27. (E)
Exercise-II
1. (B) 2. (C) 3. (E) 4. (A) 5. (D)
6. (E) 7. (B) 8. (C) 9. (A) 10. (E)
11. (C) 12. (D) 13. (A) 14. (D) 15. (B)
16. (B) 17. (B) 18. (D) 19. (C) 20. (B)
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