Let a relation \( R \) is defined on a set \( A=\{1,2,3,4,5\} \) as \( R=\{(a,b):|a-b| \) is even or zero}. Then which of the following statements is not TRUE ?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → The number of elements in \( R \) is 25.
|a − b| is even (or zero) exactly when a and b are both odd or both even. Each option is a statement; the one marked Incorrect is the statement that is not true.
Option (1) \( R \) is an equivalence relation. — Correct. R is reflexive (|a − a| = 0), symmetric (|a − b| = |b − a|) and transitive (same parity is passed on).
Option (2) The number of elements in \( R \) is 13. — Correct. Pairs from {1, 3, 5}: 3 × 3 = 9; pairs from {2, 4}: 2 × 2 = 4; total 13.
Option (3) No element of \( \{1,3,5\} \) is related to any element of \( \{2,4\} \). — Correct. An odd and an even number never have an even difference.
Option (4) The number of elements in \( R \) is 25. — Incorrect. R has 13 elements; 25 is the number of pairs in A × A.