A relation is defined on as if and only if . Then is:
A relation $R$ is defined on $\mathbb{Z}$ as $a R b$ if and only if $a^2 - 7ab + 6b^2 = 0$.
Then $R$ is:
- areflexive and symmetric
- btransitive but not reflexive
- csymmetric but not reflexive
- dreflexive but not symmetric
Answer
Answer
AIWritten by AI - it can contain mistakes.
Correct option: d
(d) reflexive but not symmetric
Reflexive: For all $a \in \mathbb{Z}$, $a^2 - 7a \cdot a + 6a^2 = a^2 - 7a^2 + 6a^2 = 0 \Rightarrow (a, a) \in R$.
Symmetric: For $(6,1)$, $6^2 - 7(6)(1) + 6(1)^2 = 36 - 42 + 6 = 0 \Rightarrow (6,1) \in R$.
However, for $(1,6)$, $1^2 - 7(1)(6) + 6(6^2) = 1 - 42 + 216 = 175 \neq 0 \Rightarrow (1,6) \notin R$.
Therefore, the relation is reflexive but not symmetric.
From ISC 2027 Specimen Mathematics Paper 1, question 1(i).