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Let R be a relation defined on N as a R b is 2a+3b is a multiple of 5,a,b∈N. Then R isnot reflexivetransitive but not symmetricsymmetric but not transitivean equivalence relation

Question

Let R be a relation defined on N as a R b is 2a+3b is a multiple of 5,a,b∈N. Then R isnot reflexivetransitive but not symmetricsymmetric but not transitivean equivalence relation

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Solution

To determine the type of relation R, we need to check if it satisfies the properties of reflexivity, symmetry, and transitivity.

  1. Reflexivity: A relation R is reflexive if for every a in N, aRa holds true. In this case, for aRa to hold true, 2a+3a = 5a should be a multiple of 5. Since 5a is always a multiple of 5 for any a in N, the relation R is reflexive.

  2. Symmetry: A relation R is symmetric if for every a, b in N, if aRb holds true, then bRa should also hold true. In this case, if 2a+3b is a multiple of 5, it doesn't necessarily mean that 2b+3a is also a multiple of 5. Therefore, the relation R is not symmetric.

  3. Transitivity: A relation R is transitive if for every a, b, c in N, if aRb and bRc hold true, then aRc should also hold true. In this case, if 2a+3b and 2b+3c are multiples of 5, it doesn't necessarily mean that 2a+3c is also a multiple of 5. Therefore, the relation R is not transitive.

So, the relation R is reflexive but not symmetric and not transitive. Therefore, it is not an equivalence relation.

This problem has been solved

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