How to Prove Equality of Sets

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

Author: bigsigma

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