Equivalence relations

Japanese school year: University year 1

What you learn

You will learn about relations satisfying reflexivity, symmetry, and transitivity, and how equivalence classes partition a set into disjoint subsets. This forms the foundation for modern algebraic concepts such as modular arithmetic and quotient structures. A solid grasp of Cartesian products and logical deductions will be advantageous.

Go to practice

Key points

This is the reflexive property of an equivalence relation $\sim$ . It means every element $a$ must always be related to (in the same group as) itself.

a \sim a

This is the symmetric property of an equivalence relation. If element $a$ is related to $b$ , then $b$ must also be related to $a$ , showing that the relation works both ways.

a \sim b \implies b \sim a

This is the transitive property of an equivalence relation. It means if $a$ relates to $b$ , and $b$ relates to $c$ , then $a$ automatically relates to $c$ .

a \sim b \land b \sim c \implies a \sim c

This defines the equivalence class $[a]$ , which gathers all elements $x$ related to $a$ within set $A$ . It is used to partition a set into non-overlapping groups.

[a] = \{ x \in A \mid x \sim a \}

Choose a set to practice.

Practice 1Coming soon