Write down all possible proper subsets each of the following sets.

(i) {1,2}

(ii) {1,2,3}

(iii) {1}

Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true?

(i) ϕ ∈ A

(ii) {ϕ} ∈ A

(iii) {1} ∈ A

(iv) {2, ϕ} ⊂ A

(v) 2 ⊂ A

(vi) {2, {1}} ⊄A

(vii) {{2}, {1}} ⊄ A

(viii) {ϕ, {ϕ}, {1, ϕ}} ⊂ A

(ix) {{ϕ}} ⊂ A.

Write down all possible subsets of each of the following sets:

Total number of the subset of any given set =2ⁿ

(i) {a}

(ii) {0,1}

(iii) {a,b,c}

(iv) {1,{1}}

(v) {ϕ}

What is the total number of proper subsets of a set consisting of n elements?

If A is any set, prove that: A ϕ A=ϕ.