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Let R be a relation on Z × Z defined by(a, b)R(c, d) if and only if ad – bc is divisible by 5. Then R isReflexive and symmetric but not transitiveReflexive but neither symmetric not transitiveReflexive, symmetric and transitiveReflexive and transitive but not symmetric

Question

Let R be a relation on Z × Z defined by(a, b)R(c, d) if and only if ad – bc is divisible by 5. Then R isReflexive and symmetric but not transitiveReflexive but neither symmetric not transitiveReflexive, symmetric and transitiveReflexive and transitive but not symmetric

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Solution

To determine the properties of the relation R, we need to check the conditions for reflexivity, symmetry, and transitivity.

  1. Reflexivity: A relation R is reflexive if for every a in Z, (a, a) R (a, a). For this relation, ad - bc = aa - aa = 0, which is divisible by 5. So, R is reflexive.

  2. Symmetry: A relation R is symmetric if for every a, b in Z, if (a, b) R (c, d) then (c, d) R (a, b). For this relation, if ad - bc is divisible by 5, then cd - da = -(ad - bc) is also divisible by 5. So, R is symmetric.

  3. Transitivity: A relation R is transitive if for every a, b, c in Z, if (a, b) R (c, d) and (c, d) R (e, f) then (a, b) R (e, f). For this relation, if ad - bc and cf - de are divisible by 5, it does not necessarily mean that af - be is divisible by 5. So, R is not transitive.

Therefore, the relation R is reflexive and symmetric but not transitive.

This problem has been solved

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