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 BC, 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: For example, let A = {1, 2} and

B = (2, 3, {1, 2}}. As 1∈ A and A∈ B but 1∉ B 

(ii) False: For example, let A = {1, 2}, B = {1, 2, 3} and C = {3, 4,{1, 2, 3}}

Now clearly, A ⊂ B and B⊂C but A ∉ C, 

(iii) True: Given A ⊂ B and 

Now we have to prove A ⊂ C 

∴ x ∈ C ∵ B ∈ C. 

∀ X ∈ A ⇒X ∈ C∴ A ∈ C. 

(iv) False: For example, let 

A = (1, 2), B = {2, 3),C = (1, 2, 4) 

Clearly A⊄B and B⊄C but A ⊂ C 

(v) False: For example, let A=(l,2), B={2, 3, 4, 5} 

Now 1 ∈ A and A ∉ B but 1 ∉ B 

(vi) True: Suppose x∈ A then X∈ B A⊂ B.