**Theorem 1: **For any three sets , prove that:

i)

(ii) .

**Proof:**

(i) Let be an arbitrary element of . Then

[By Definition]

[By Definition of Union]

… … … … … i)

Now, let be an arbitrary element of .

Then,

… … … … … ii)

Hence from i) and ii) we get

ii) Let be an arbitrary element of . Then,

… … … … … i)

Let be an arbitrary element of . Then,

… … … … … ii)

Hence, from (i) and (ii), we get .

**Theorem 2:** For any three sets , prove that: .

**Proof:**

Let be an arbitrary element of . Then

… … … … … i)

Again, let be an arbitrary element of . Then,

… … … … … ii)

Hence, from (i) and (ii), we get

**Theorem 3:** If and are any two non-empty sets, then prove that: .

**Proof:**

First, let . Then we have to prove that .

and,

Conversely, let . Then we have to Prove that .

Let be an arbitrary element of . Then,

for all

Again, let be an arbitrary element of . Then,

for all

Hence, .

**Theorem 4:** If , show that .

**Proof:** Let be an arbitrary element of

Then,

Hence,

**Theorem 5:** If , prove that: for any set .

**Proof: ** Let be an arbitrary element of .

Then,

**Theorem 6: **If and , prove that:

**Proof: **Let be an arbitrary element of ,

Then

**Theorem 7: **For any sets prove that: .

**Proof:**

Let be an arbitrary element of . Then,

Similarly,

Hence,

**Theorem 8: **For any three sets prove that:

(i)

ii)

**Proof:**

i) [ By De-Morgan’s law ]

ii) [ By De-Morgan’s law ]

**Theorem 9: **Let and be two non-empty sets having elements in common, then prove that and have elements in common.

**Proof:**

We have,

[On replacing by and by ]

It is given that latex n elements, so has elements.

But

has elements

Hence have elements in common

**Theorem 10: **Let be a non-empty set such that . Show that .

**Proof:**

Let be an arbitrary element of .

Then, for all

for all

Thus,

Let be an arbitrary element of .

Then, for all

for all

From i) and ii) we get .