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?

  • 1
    $\begingroup$ The second is correct. Being reflexive means that all elements must be related to themselves, not just some. If there is even one example of an element which is not related to itself, then it is not reflexive. $\endgroup$ – JMoravitz Oct 16 '17 at 3:03
  • $\begingroup$ Ok. That makes sense. Thank you. $\endgroup$ – ErinA Oct 16 '17 at 3:08
  • $\begingroup$ @ErinA : See my edits to the question for proper MathJax usage. $\endgroup$ – Michael Hardy Oct 16 '17 at 4:06

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 $x\in A,\ x\mathbin{R} x.$


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