Indicate which of the following relations on the given sets are reflexive on the given set, which are symmetric, and which are transitive.
$$A = \{(x, y) \in \Bbb{Z} \times \Bbb{Z}: x + y = 10\}$$
I'm just a tad confused about the reflexivity of this relation.
I'm inclined to say the relation is reflexive. By letting $x=5$, we can see that $x + x = 10$ is true. I'm choosing to let $x=5$ because, by definition of reflexivity, $\forall x\in A,\ xRx$. By definition, $x$ must equal $x$ and the only number that could make this relation true is 5.
However, I came across this answer: Consider the relation, $A =\{(x, y) \in \Bbb{Z} \times \Bbb{Z}: x + y = 10\}$
Let $(x ,y)= (3, 7)$. It follows that $3 + 7= 10$. Therefore, $(3, 7) \in A$. But $3 + 3=6$ and $3 + 3 \ne 10$. Therefore $(3, 3) \notin A$. Thus the relation is not reflexive on the set
Can someone explain which answer is right, and why?
If the given set is $\mathbb Z,$ then this relation is not reflexive on that set, because, as you have pointed out, there are counterexamples.
That a relation $R$ is reflexive on a set $A$ means that for every $x\in A$ you have $x\mathbin{R}x.$
It does not mean that for some $x\in A,\ x\mathbin{R} x.$