Sets – Venn diagrams and Operations on Set (Exercises)

Q.1. (A∪B)-C = (A-C) ∪ (B-C)

Sol. let x ∈ [(A∪B)-C]

x ∈ (A∪B) and x ∉ C

(x ∈ A or x ∈ B) and x ∉ C

(x ∈ A and x ∉ C) or (x ∈ B and x ∉ C)

x ∈ {(A-C) or ∈ (B-C)}

x ∈ {(A-C) ∪ (B-C)}

(A∪B)-C (A-C) ∪ (B-C) --- i

Again, let y ∈ (A-C) ∪ (B-C)

y ∈ (A-C) or y ∈ (B-C)

(y ∈ A and y ∉ C) or (y ∈ B and y ∉ C)

(y ∈ A or y ∈ B) and y ∉ C

y ∈ (A∪B) and y ∉ C

y ∈ {(A∪B)-C}

(A-C) ∪ (B-C) (A∪B)-C ---ii

From equation (i) and (ii),

∴ (A∪B)-C = (A-C) ∪ (B-C)

Q.2. A-(B∪C) = (A-B) ∩ (A-C)

Sol. let x ∈ {A-(B∪C)}

x ∈ A and x ∉ (B∪C)

x ∈ A and (x ∉ B and x ∉ C)

(x ∉ A and x ∉ B) and (x ∉ A and x ∉ C)

x ∈ (A-B) and x ∈ (A-C)

x ∈ {(A-B) ∩ (A-C)}

A- (A-B) (A-B) ∩ (A-C) ---i

Again, let

y ∈ (A-B) ∩ (A-C)

y ∈ (A-B) and y ∈ (A-C)

(y ∉ A and y ∉ B) and (y ∉ A and y ∉ C)

y ∈ A and y ∉ B∪C

y ∈ { A-(B∪C)}

(A-B) ∩ (A-C) A-(B∪C) -- ii

From equation (i) and (ii),

A-(B∪C) (A-B) ∩ (A-C)

Q.3. A∩(B-C) = (A∩B) - (A∩C)

Sol. let x ∈ {A∩ (B-C)}

x ∈ A and x ∈ B-C

x ∈ A and x ∈ B and x ∉ C

(x ∈ A and x ∈ B) and (x ∈ A and x ∉ C)

x ∈ (A∩B) and x ∉ (A∩C)

x ∈ {(A∩B) - (A∩C)}

A∩ (B∩C) (A∩B) - (A∩C) --- i

Again, let

y ∈ (A∩B) ∩ (A-C)

y ∈ A and y ∈ B and y ∉ C

y ∈ A and y ∈ B-C

y ∉ { A∩ (B-C)}

(A∩B)-(A∩C) A ∩ (B-C) --- ii

From equation (i) and (ii),

A∩ (B-C) = (A∩B) - (A∩C)

Q.4. For any two sets A and B, prove that A∪B = A∩BA=B

Sol. let A =B, then A∪B=A and A∩B=A

A∪B = A∩B

Thus, A=B --- i

Conversely, let A∪B = A∩B

Now, let x ∈ A

x ∈ (A∪B) ∴[ A∪B = A∩B]

x ∈ (A∩B)

x ∈ A and x ∈ B

x ∈ B

A B ---- ii

Now, let y ∈ A

y ∈ A∪B ∴[ A∪B = A∩B]

y ∈ A∩B

y ∈ A and y ∈ B

y ∈ A

B A ---- iii

From eqs ii and iii, We get A=B

Thus (A∪B) = (A∩B)

A=B

From eqs iii and iv, We get

A∪B = A∩B ⇔ A=B