Question

Let R be a relation on the set of all integers such that aRb if

and only if a + b is even. prove if R is reflexive,
symmetric, anti-symmetric, and transitive.

Answer 1

To prove if the relation R is reflexive, symmetric, anti-symmetric, and transitive, we need to analyze each property. The relation R is reflexive because every element is related to itself. The relation R is symmetric because if a is related to b, then b is related to a.

Step-by-step explanation:

In order to prove if the relation R is reflexive, symmetric, anti-symmetric, and transitive, we need to analyze each property:

Reflexive Property:

A relation is reflexive if every element is related to itself. In this case, for any integer a, we have aR a if a + a is even. Since a + a is always even, the relation R is reflexive.

Symmetric Property:

A relation is symmetric if whenever a is related to b, then b is related to a. In this case, if aR b, then a + b is even. Similarly, bR a if b + a is even, which is the same condition. Therefore, R is symmetric.

Anti-Symmetric Property:

A relation is anti-symmetric if whenever aR b and bR a, then a = b. In this case, if aR b and bR a, then a + b is even and b + a is even, which implies that a = b. Hence, R is anti-symmetric.

Transitive Property:

A relation is transitive if whenever aR b and bR c, then aR c. In this case, if aR b and bR c, then a + b is even and b + c is even. Adding these two equations, we get (a + b) + (b + c) = a + (b + b) + c = a + 2b + c = (a + c) + 2b, which is even. Therefore, aR c and R is transitive.