• Prove some fundamentals laws of algebra of sets :

1. A U A=A
2. A intersection A=A
3. Identity laws for any set A
4. Commutative laws for any sets A & B
5. Associative laws, A,B&C sets
6. DISTRIBUTIVE LAWS, A,B&C SETS
7. DE MORGANS LAW, for any sets A & B

(1) A ∪ A = A

Let A = {x : x ∈ A}

A ∪ A = {x : x ∈ a or x ∈ A}

 = {x : x ∈ A}

 = A

(2) Identity law, A ∪ Ø = A

Let A = {x : x ∈ A}

A ∪ Ø = {x : x ∈ A or x ∈ Ø}

 = {x : x ∈ A}

 = A

(3) Commutative law, A ∪ B = B ∪ A

Let x ∈ A ∪ B

Then, x ∈ A or x ∈ B

⇒ x ∈ B or x ∈ A

⇒ x ∈ B ∪ A

⇒ A ∪ B ⊆ B ∪ A

Similarly, B ∪ A ⊆ A ∪ B

⇒ A ∪ B = B ∪ A

(4) De-morgan's law, (A ∪ B)' = A' ∩ B'

Let x ∈ (A ∪ B)'

⇒ x (A ∪ B)

⇒ x A  and x   B

⇒ x ∈ A' and x ∈ B'

⇒ x ∈ A' ∩ B'

⇒ (A ∪ B)' ⊆ A' ∩ B'

Now, let y ∈ A' ∩ B'

⇒ y ∈ A' and y ∈ B'

⇒ y A and y B

⇒ y A ∪ B

⇒ y ∈ (A ∪ B)'

⇒ A' ∩ B' ⊆ (A ∪ B)'

Hence,(A ∪ B)' =  A' ∩ B'

Similarly, you can prove other laws also.