Question

prove that
A-(B⋂C) = (A-B) ∪ (A-C)​

Answer 1

Answer:

A - ( B ∩ C ) = ( A - B ) ∪ ( A - C )

Step-by-step-explanation:

Let any three sets as A, B and C.

A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

B = { 2, 4, 6, 8, 10 }

C = { 2, 3, 5, 7 }

Now,

LHS = A - ( B ∩ C )

∴ We have to find B ∩ C first.

B = { 2, 4, 6, 8, 10 }

C = { 2, 3, 5, 7 }

∴ B ∩ C = { 2 }

Now,

A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

B ∩ C = { 2 }

∴ A - ( B ∩ C ) = { 1, 3, 4, 5, 6, 7, 8, 9, 10 } - - ( 1 )

Now,

RHS = ( A - B ) ∪ ( A - C )

∴ We need to find ( A - B ) & ( A - C ) first.

Now,

A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

B = { 2, 4, 6, 8, 10 }

∴ A - B = { 1, 3, 5, 7, 9 }

Now,

A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

C = { 2, 3, 5, 7 }

∴ A - C = { 1, 4, 6, 8, 9, 10 }

Now,

A - B = { 1, 3, 5, 7, 9 }

A - C = { 1, 4, 6, 8, 9, 10 }

∴ ( A - B ) ∪ ( A - C ) = { 1, 3, 4, 5, 6, 7, 8, 9, 10 } - - ( 2 )

From ( 1 ) & ( 2 )

A - ( B ∩ C ) = ( A - B ) ∪ ( A - C )

Hence proved!

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Additional Information:

1. Set:

The group of well defined object is known as 'set'.

Ex.: A set of days in a week.

2. Types of set:

There are total four types of sets.

A. Singleton set

B. Empty set or Null set

C. Finite set

D. Infinite set

3. Methods of writing a set:

A. Listing method:

X = { 1, 3, 5, 7, 9 }

B. Roster form or Rule method:

X = { x | x is an odd number, x ∈ N, x ≤ 1 }

4. Union of two sets:

The set formed by the union of all elements in two sets is called Union of two sets.

5. Intersection of two sets:

The set formed by only common elements in two sets is called intersection of two sets.

6. Addition of two sets:

The union of all elements in two sets is called the addition of two sets. It is same as the Union of two sets.

7. Subtraction of two sets:

The set formed by removing common elements from one of the two sets is called subtraction of two sets.

Answer 2

Answer:

Step-by-step explanation:  For any three sets A, B and C, we are given to prove the following :

A - (B ∩ C) = (A - B) U (A - C).

We know that

any two sets P and Q are equal if and only if both are subsets of each other, that is

P ⊂ Q and Q ⊂ P.

Let us consider that

  x ∈ A - (B ∩ C)

⇒x∈A, x ∉ (B ∩ C)

⇒x∈A, (x∉B or x∉C)

⇒(x∈A, x∉B) or (x∈A, x∉C)

⇒x∈(A-B) or x∈(A-C)

⇒x∈(A-B) ∪ (A-C)

So, A - (B n C) ⊂ (A-B) U (A-C).

Again, let

 x∈(A-B) ∪ (A-C)

⇒x∈(A-B) or x∈(A-C)

⇒(x∈A, x∉B) or (x∈A, x∉C)

⇒x∈A, (x∉B or x∉C)

⇒x∈A, x ∉ (B ∩ C)

⇒x ∈ A - (B ∩ C).

So, A - (B ∩ C) ⊂ (A-B) ∪ (A-C).

Therefore, we get

A - (B n C) = (A - B) U (A - C).

Hence proved

Step-by-step explanation: