What is an Equivalence Relation? | Reflexive, Symmetric, and Transitive Properties

What are equivalence relations? Equivalence relations are very nice relations to work with, and we are very familiar with a particular equivalence relation: equals! It is at times important to be able to recognize other equivalence relations, so we discuss what they are in today's math video lesson!
An equivalence relation is a binary relation that has three particular properties. A binary relation is basically a relation that relates two objects at a time, most of the relations we are familiar with are binary relations: like equal to and less than.
The three properties a binary relation has to have to be an equivalence relation are the reflexive, symmetric, and transitive properties.
We will call our relation R. The relation R is reflexive if every object under consideration relates to itself. The relation R is symmetric if it is true, for all objects under consideration, that when a relates to b, b also relates to a. The relation R is transitive if it is true, for all objects under consideration, that when a relates to b, and b relates to c, it is also true that a relates to c.
As an example of the reflexive property, 4 = 4. As an example of the symmetric property, 4 = 2*2 and 2*2 = 4. As an example of the transitive property, 8 = 4*2 and 4*2 = 2*2*2, and it is true that 8 = 2*2*2.
A relation that does not have some of these properties is less than on the real numbers. Notice that 3 is not less than 3, so it is not a reflexive relation. Also, 3 is less than 4, but 4 is not less than 3, so it is not a symmetric relation.
An example of a relation that breaks transitivity is the relation "is the father of". If Starbuck is the father of Stubb, and Stubb is the father of Flask, it is not true that Starbuck is the father of Flask, thus the relation is not transitive.
For a more mathematical example, consider the relation "set membership" or "is an element of". The number 2 is an element of the set { 1, 2 }. The set { 1, 2 } is an element of the set { { 1, 2 }, 4 }. But 2 is not an element of the set { {1, 2}, 4 }
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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