**Proofs for De Morgan's laws :**

Here we are going to see the proof of De morgan's laws by Venn diagram.

**De morgan's law for set difference :**

For any three sets A, B and C, we have

**(i) A \ (B u C) = (A \ B) n (A \ C)**

**(ii) A \ (B n C) = (A \ B) u (A \ C)**

**De morgan's law for set complementation :**

Let U be the universal set containing sets A and B. Then

**(i) (A u B)' = A' n B'**

**(ii) (A n B)' = A' u B'**

**A \ (B n C) = (A \ B) u (A \ C)**

From the above Venn diagrams (2) and (5), it is clear that

A \ (B n C) = (A \ B) u (A \ C)

**Hence, De morgan's law for set difference is verified.**

Now, let us look at the Venn diagram proof of De morgan's law for complementation.

**(A n B)' = A' u B'**

From the above Venn diagrams (2) and (5), it is clear that

(A n B)' = A' u B'

**Hence, De morgan's law for complementation is verified.**

**Problem 1 : **

Let A = { a, b, c, d, e, f, g, x, y, z }, B = { 1, 2, c, d, e } and C = { d, e, f, g, 2, y }. Verify De Morgan’s laws of set difference.

**Solution : **

First, we shall verify A \ (B u C) = (A \ B) n (A \ C)

To do this, we consider

B u C = { 1, 2, c, d, e } u { d, e, f, g, 2, y }

B u C = { 1, 2, c, d, e, f, g, y }

We know that

A / (B u C) = { a, b, c, d, e, f, g, x, y, z } \ { 1, 2, c, d, e, f, g, y }

A / (B u C) = { a, b, x, z } ---------(1)

A \ B = { a, b, f, g, x, y, z }

A \ C = { a, b, c, x, z }

(A \ B) n (A \ C) = { a, b, x, z } ---------(2)

From (1) and (2), it is clear that A \ (B u C) = (A \ B) n (A \ C)

Similarly, one can verify A \ (B n C) = (A \ B) u (A \ C) for the given sets above.

**Problem 2 : **

Let U = { - 2, -1, 0, 1, 2, 3, 4, 5, .......10 }, A = {- 2, 2, 3, 4, 5 } and B = { 1, 3, 5, 8, 9 }. Verify De Morgan’s laws of complementation

**Solution : **

First, we shall verify (A u B)' = A' n B'

To do this, we consider

A u B = {- 2, 2, 3, 4, 5 } u { 1, 3, 5, 8, 9 }

A u B = { -2, 1, 2, 3, 4, 5, 8, 9 }

We know that

(A u B)' = U \ { -2, 1, 2, 3, 4, 5, 8, 9 }

(A u B)' = { -1, 0, 6, 7, 10 } ---------(1)

A' = U \ A = U \ { -2, 2, 3, 4, 5 } = { -1, 0, 1, 6, 7, 8, 9, 10 }

B' = U \ B = U \ { 1, 3, 5, 8, 9 } = { -2, -1, 0, 2, 4, 6, 7, 10 }

A' n B' = { -1, 0, 1, 6, 7, 8, 9, 10 } n { -2, -1, 0, 2, 4, 6, 7, 10 }

A' n B' = { -1, 0, 6, 7, 10 } ---------(2)

From (1) and (2), it is clear that (A u B)' = A' n B'

Similarly, one can verify (A n B)' = A' u B' for the given sets above.

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