Relations and Equivalence Relations

Videos (~50 minutes)

Equivalence Relations (24:08)
Proving Equivalence Relations (22:40)

Reading (~30 minutes)

Book of Proof, Sections 11.1-11.4.

Warmup (~40 minutes)

Problem 1

Let \(A = \{a, b, c, d, e\}\). Suppose that \(R\) is an equivalence relation on \(A\). Suppose further that \(R\) has two equivalence classes, and that \(aRd\), \(bRc\), and \(eRd\). Fully describe \(R\) by either writing it as a set or drawing it.

Problem 2

For each of the three relations below:

State whether it is an equivalence relation or not.
If that relation is an equivalence relation, write a short proof including the correct vocabulary for all the properties that an equivalence relation must satisfy.
If not, demonstrate a property of equivalence relations fails to be satisfied by showing a counterexample. It is sufficient to demonstrate that one property does not hold, even if there are multiple. A single counterexample is enough!

For each of the relations below, the domain is the integers \(\mathbb{Z}\).

Part A

The relation \(aRb\) means that \(a - b\) is an even number.

Part B

The relation \(aRb\) means that \(a - b > 2\).

Part C

The relation \(aRb\) means that \(a - b\) is an odd number.