**Proof by Venn diagram :**

Here we are going to see the proof of the following properties of sets operations and De morgan's laws by Venn diagram.

The following are the important properties of set operations.

**(i) COMMUTATIVE PROPERTY **

(a) A u B = B u A (Set union is commutative)

(b) A n B = B n A (Set intersection is commutative)

**(ii) ASSOCIATIVE PROPERTY**

(a) A u (B u C) = (A u B) u C

(Set union is associative)

(b) A n (B n C) = (A n B) n C

(Set intersection is associative)

**(iii) DISTRIBUTIVE PROPERTY**

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

(Intersection distributes over union)

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

(Union distributes over intersection)

**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'**

**Problem 1 :**

For the given sets A = { -10, 0, 1, 9, 2, 4, 5 } and B = {-1, -2, 5, 6, 2, 3, 4 }, verify that

(i) Set union is commutative. Also verify it by using Venn diagram.

(ii) Set intersection is commutative. Also verify it by using Venn diagram.

**Solution : **

(i) Let us verify that union is commutative.

A u B = { -10, 0, 1, 9, 2, 4, 5 } u {-1, -2, 5, 6, 2, 3, 4 }

A u B = { -10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9 } ---------(1)

B u A = {-1, -2, 5, 6, 2, 3, 4 } u { -10, 0, 1, 9, 2, 4, 5 }

B u A = { -10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9 } ---------(2)

From (1) and (2), we have

A u B = B u A

By Venn diagram, we have

From the above two Venn diagrams, it is clear that

A u B = B u A

**Hence, it is verified that set union is commutative.**

(ii) Let us verify that union is commutative.

A n B = { -10, 0, 1, 9, 2, 4, 5 } n {-1, -2, 5, 6, 2, 3, 4 }

A n B = { 2, 4, 5 } ---------(1)

B n A = {-1, -2, 5, 6, 2, 3, 4 } u { -10, 0, 1, 9, 2, 4, 5 }

B n A = { 2, 4, 5 } ---------(2)

From (1) and (2), we have

A n B = B n A

By Venn diagram, we have

From the above two Venn diagrams, it is clear that

A n B = B n A

**Hence, it is verified that set intersection is commutative.**

Let us look at the next problem on "Proof by Venn diagram"

**Problem 2 :**

For the given sets A = { 1, 2, 3, 4, 5 }, B = { 3, 4, 5, 6 } and C = { 5, 6, 7, 8 }, verify that A u (B u C ) = (A u B) u C. Also verify it by using Venn diagram.

**Solution : **

Let us verify that set union is associative.

B u C = { 3, 4, 5, 6 } u { 5, 6, 7, 8 }

B u C = { 3, 4, 5, 6, 7, 8 }

A u (B u C) = { 1, 2, 3, 4, 5 } u { 3, 4, 5, 6, 7, 8 }

A u (B u C) = { 1, 2, 3, 4, 5, 6, 7, 8 } ---------(1)

A u B = { 1, 2, 3, 4, 5 } u { 3, 4, 5, 6 }

A u B = { 1, 2, 3, 4, 5, 6 }

(A u B) u C = { 1, 2, 3, 4, 5, 6 } u { 5, 6, 7, 8 }

(A u B) u C = { 1, 2, 3, 4, 5, 6, 7, 8 } ---------(2)

From (1) and (2), we have

A u (B u C) = (A u B) u C

By Venn diagram, we have

From the above Venn diagrams (2) and (4), it is clear that

A u (B u C) = (A u B) u C

**Hence, it is verified that set union is associative.**

Let us look at the next problem on "Proof by Venn diagram"

**Problem 3 :**

For the given sets A = { a, b, c, d }, B = { a, c, e } and C = { a, e }, verify that A n (B n C) = (A n B) n C. Also verify it by using Venn diagram.

**Solution : **

Let us verify that set intersection is associative.

B n C = { a, c, e } u { a, e }

B n C = { a, e }

A n (B n C) = { a, b, c, d } n { a, e }

A n (B n C) = { a } ---------(1)

A n B = { a, b, c, d } u { a, c, e }

A n B = { a, c }

(A n B) n C = { a, c } n { a, e }

(A n B) n C = { a } ---------(2)

From (1) and (2), we have

A n (B n C) = (A n B) n C

By Venn diagram, we have

From the above Venn diagrams (2) and (4), it is clear that

A n (B n C) = (A n B) n C

**Hence, it is verified that set intersection is associative.**

Let us look at the next problem on "Proof by Venn diagram"

**Problem 4 :**

For the given sets A = { 0, 1, 2, 3, 4 }, B = { 1, -2, 3, 4, 5, 6 } and C = { 2, 4, 6, 7 }, verify that A u (B n C ) = (A u B) n (A u C). Also verify it by using Venn diagram.

**Solution : **

Let us verify that union distributes over intersection.

B n C = { 1, -2, 3, 4, 5, 6 } n { 2, 4, 6, 7 }

B n C = { 4, 6 }

A u (B n C) = { 0, 1, 2, 3, 4 } u { 4, 6 }

A u (B n C) = { 0, 1, 2, 3, 4, 6 } ---------(1)

A u B = { 0, 1, 2, 3, 4 } u { 1, -2, 3, 4, 5, 6 }

A u B = { -2, 0, 1, 2, 3, 4, 5, 6 }

A u C = { 0, 1, 2, 3, 4 } u { 2, 4, 6, 7 }

A u C = { 0, 1, 2, 3, 4, 6, 7 }

(A u B) n (A u C) = { -2, 0, 1, 2, 3, 4, 5, 6 } n { 0, 1, 2, 3, 4, 6, 7 }

(A u B) n (A u C) = { 0, 1, 2, 3, 4, 6 } ---------(1)

From (1) and (2), we have

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

By Venn diagram, we have

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

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

**Hence, it is verified that union distributes over intersection.**

**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.**

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