Let A = {a, b, c}, B = {b, d, e} and C = {e, f, g}, verify that :

(i) A ∪ B = B ∪ A

(ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)

(iii) A ∩ B = B ∩ A

(iv) (A ∩ B) ∩ C = A ∩ (B ∩ C)

(i) We have:

A = {a, b, c}

B = {b, d, e}

A ∪ B = {a, b, c} ∪ {b, d, e} = {a, b, c, d, e} $\quad$…..(1)

B ∪ A = {b, d, e} ∪ {a, b, c} = {a, b, c, d, e} $\quad$……(2)

Since, (1) and (2) are equal,

∴ A ∪ B = B ∪ A

(ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)

We have:

A = {a, b, c}

B = {b, d, e}

C = {e, f, g}

A ∪ B = {a, b, c} ∪ {b, d, e} = {a, b, c, d, e}

(A ∪ B) ∪ C = {a, b, c, d, e} ∪ {e, f, g} = {a, b, c, d, e, f, g} $\quad$…..(1)

Again, B ∪ C = {b, d, e} ∪ {e, f, g} = {b, d, e, f, g}

A ∪ (B ∪ C) = {a, b, c} ∪ {b, d, e, f, g} = {a, b, c, d, e, f, g} $\quad$……(2)

Since, (1) and (2) are equal,

∴ (A ∪ B) ∪ C = A ∪ (B ∪ C)

(iii) A ∩ B = B ∩ A

We have:

A = {a, b, c}

B = {b, d, e}

A ∩ B = {a, b, c} ∩ {b, d, e} = {b} $\quad$…..(1)

B ∩ A = {b, d, e} ∩ {a, b, c} = {b} $\quad$……(2)

Since, (1) and (2) are equal,

∴ A ∩ B = B ∩ A

(iv) (A ∩ B) ∩ C = A ∩ (B ∩ C)

We have:

A = {a, b, c}

B = {b, d, e}

C = {e, f, g}

A ∩ B = {a, b, c} ∩ {b, d, e} = {b}

(A ∩ B) ∩ C = {b} ∩ {e, f, g} = { } or ϕ $\quad$…..(1)

B ∩ C = {b, d, e} ∩ {e, f, g} = {e}

A ∩ (B ∩ C) = {a, b, c} ∩ {e} = { } or ϕ $\quad$……(2)

Since, (1) and (2) are equal,

∴ (A ∩ B) ∩ C = A ∩ (B ∩ C)