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Let $x$ and $y$ be integers. Determine whether the following relations are reflexive, symmetric, antisymmetric, or transitive.

  1. $x \equiv y \mod{7}$
  2. $xy \geq 1$
  3. $x = y^2$

So far in class we have only determined whether sets are reflexive, symmetric, antisymmetric, or transitive. How should I approach this when given problems like such?

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    These relations was just as well be viewed as the sets of all pairs $(x, y)$ that satisfy them. For example (a) defines the set $\{(x, y) \in \Bbb Z \times \Bbb Z : x \equiv y \pmod 7\}$. – 2017-02-06
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    You can easily check the following : if $xRy$, then $(x,y)\in R$ – 2017-02-06
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    Please read the entry for [Equivalence Relation](https://en.wikipedia.org/wiki/Equivalence_relation), especially sections 1 - 4 (see top outline). (In the fourth section, you find information on relations which require anti-symmetry (namely, a partial order). Once you've read those sections (not more than a page) C'mon back and summarize the definitions in you question post. We can help you then apply the definitions to the example problem(s). – 2017-02-06

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