# How to Prove Equality of Sets

In this tutorial we see how to prove equality of sets. We say that Set A is equal to set B if and only if A is a subset of B and B is a subset of A.

## How to Prove Equality of Sets – method 1

Using the above definition of equality of sets, It is straight forward way to prove the equality of set. To prove that A=B, we need to show that A⊆B and B⊆A.

As we know, A is a subset of B if and only if each item of A also belongs to B. In other words, to prove the equality of sets we need to show:

• Each item of A belongs to B – Suppose x∈A , we need to that x∈B
• Each item of B belongs to A – Suppose x∈B , we need to that x∈A

## How to Prove Equality of Sets – method 2

If we can show that x∈A if and only if x∈B. We prove that A=B. Let’s see why:

• x∈A implies that x∈B – This means that A⊆B
• x∈B implies that x∈A – This means that B⊆A

Since A⊆B and B⊆A, we prove that A=B.

## How to Prove Equality of Sets – method 3

When solving an equation, we sometimes use well-known identities such (x+y)2=x2+2xy+y2. We do not prove the well-known identities, we only quote those identities. In other words, It is enough to mention well-known identities when solving the equation.

We can also use well-known identities, when proving the equality of sets. For example, we can use the identity (Ac)c=A.. As we do not need to prove the identity (x+y)2=x2+2xy+y2 , we do not need to prove this identities. We can rely on the work of others.

## How to Prove Equality of Sets in action

In the next movie, we see the above techniques in action:

### Table of content of the movie:

0:10 – How to Prove Equality of Sets with method 1
1:11 – How to Prove Equality of Sets with method 2
2:12 – How to Prove Equality of Sets with method 3 