Let A be a set with 3 elements. Consider the following statements:
(A) The number of relations on A is 512.
(B) The number of reflexive relations on A is 64.
(C) The number of symmetric relations on A is 32.
(D) The number of one-one functions from A to A is 9.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
A relation on A is a subset of \(A \times A\), which has \(3 \times 3 = 9\) pairs:
number of relations \(= 2^{9} = 512\)
A reflexive relation must contain the 3 pairs (a, a); each of the other \(9 - 3 = 6\) pairs is free:
number of reflexive relations \(= 2^{6} = 64\)
A symmetric relation is fixed by 3 free choices for the pairs (a, a) and 3 free choices for the unordered pairs {a, b} with \(a \ne b\):
number of symmetric relations \(= 2^{3 + 3} = 2^{6} = 64\)
A one-one function from A to A uses each element once as an image:
\(3 \times 2 \times 1 = 6\)
So A has 512 relations, 64 reflexive relations, 64 symmetric relations and 6 one-one functions to itself.
Checking the statements
(A) The number of relations on A is 512. — Correct. \(2^{9} = 512\).
(B) The number of reflexive relations on A is 64. — Correct. \(2^{6} = 64\).
(C) The number of symmetric relations on A is 32. — Incorrect. The number is \(2^{6} = 64\).
(D) The number of one-one functions from A to A is 9. — Incorrect. The number is \(3! = 6\).