Solutions to Problem Set 22

Math 310-01/02

11-10-2017

1. A relation $\sim$ is defined on $\real^2$ by

$$(a, b) \sim (c, d) \quad\hbox{if and only if}\quad a b c d \ge 0.$$

Check each axiom for an equivalence relation. If the axiom holds, prove it. If the axiom doesn't hold, give a specific counterexample.

If $(a, b) \in \real^2$ , then

$$a b a b = a^2 b^2 \ge 0.$$

Hence, $(a, b) \sim (a, b)$ . The relation is reflexive.

If $(a, b), (c, d) \in \real^2$ and $(a, b) \sim (c, d)$ , then

$$a b c d \ge 0.$$

Hence,

$$c d a b \ge 0.$$

Therefore,

$$(c, d) \sim (a, b).$$

The relation is symmetric.

Note that $(1, -1) \sim (0, 0)$ and $(0, 0) \sim (1, 1)$ , because

$$1 \cdot (-1) \cdot 0 \cdot 0 = 0 \ge 0 \quad\hbox{and}\quad 1 \cdot 0 \cdot 1 \cdot 1 = 0 \ge 0.$$

But

$$1 \cdot (-1) \cdot 1 \cdot 1 = -1 \not\ge 0.$$

Hence, $(1, -1) \not\sim (1,
   1)$ . The relation is not transitive.


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