Question

In each of the following, determine whether the statement is true or false. 

If it is true, prove it. If it is false, give an example. 

(i) If x ∈ A and A ∈ B, then x ∈ B 

(ii) If A ⊂ B and B ∈ C, then A ∈ C 

(iii) If A ⊂ B and B ⊂ C, then A ⊂ C 

(iv) If A ⊄ B and B ⊄ C, then A ⊄ C

(v) If x ∈ A and A ⊄ B, then x ∈ B 

(vi) If A ⊂ B and x ∉ B, then x ∉ A 

Answer 1

(i) False 

Let 

A = {1, 2} and B = {1, {1, 2}, {3}} 

(vi) True

 Let 

A ⊂ B and x ∉ B. 

To show: x ∉ A If possible, suppose x ∈ A. 

Then, x ∈ B, 

which is a contradiction as x ∉ B ∴x ∉ A