Answer:
Let's first define what it means for a relation to be an equivalence relation or a partial order (poset):
Equivalence relation: A relation on a set is an equivalence relation if it is reflexive, symmetric, and transitive. That is, for all a, b, and c in the set:
Reflexivity: aRa (a is related to itself)
Symmetry: If aRb then bRa (if a is related to b, then b is related to a)
Transitivity: If aRb and bRc, then aRc (if a is related to b and b is related to c, then a is related to c)
Partial order (poset): A relation on a set is a partial order if it is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in the set:
Reflexivity: aRa
Antisymmetry: If aRb and bRa, then a = b (if a is related to b and b is related to a, then a and b are equal)
Transitivity: If aRb and bRc, then aRc
Now let's apply these definitions to the two relations given:
(i) aRb if and only if a = b or a - b is even
Reflexivity: aRa since a = a or a - a = 0 (which is even)
Symmetry: If aRb, then either a = b or a - b is even. If a = b, then bRa since b = a or b - a = 0 (which is even). If a - b is even, then b - a is also even, so bRa. Therefore, the relation is symmetric.
Transitivity: If aRb and bRc, we have two cases to consider:
If a = b and b = c, then a = c and aRc.
If a - b and b - c are both even, then a - c is even (the sum of two even numbers is even), so aRc.
If a - b and b - c are both odd, then a - c is even (the sum of two odd numbers is even), so aRc. Therefore, the relation is transitive.
Thus, we can conclude that relation (i) is an equivalence relation
Explanation: