Question

State whether the given statement is true false:

(i) If A ⊂ B and x∉B than x ∉A.

(ii) If A ⊆ϕ then A = ϕ

(iii) If A, B and C are three sets such than A ϵ B and B ⊂ C then A ⊂ C.

(iv) If A, B and C are three sets such than A ⊂B and B ϵ C then A ϵC.

(v) If A, B and C are three sets such that A ⊄B and B ⊄C then A ⊄C.

(vi) If A and B are sets such that x A and A ϵ B then x ϵ B.

Answer 1

(i) True 

Explanation: We have A ⊂ B since A is a subset of B then all elements of A should be in B. 

Let A = {1,2} and B = {1,2,3} 

Let x=4∉B 

Also we observe that 4∉A. 

Hence, If A ⊂ B and x∉B than x ∉A. 

(ii) True

Explanation: We have, A ⊆ ϕ

Now, A is a subset of null set , this implies A is also an empty set.

⇒ A =ϕ 

(iii) False 

Explanation: Let A = {a}, B = {{a}, b}

here , A ϵ B

Now, let C = {{a}, b, c}. 

Since, {a},b is in B and also in C thus, B ⊂ C. 

But, A ={a} and {a} is an element of C, since the element of a set cannot be a subset of a set. 

Hence,A ⊄ C. 

(iv) False

Explanation: Let A = {a},B = {a, b} and C = {{a, b}, c}. 

Then,A⊂ B and B ϵC. But, A ∉ C since {a} is not an element of C. 

(v) False. 

Explanation: Let A = {a}, B = {b, c} and C = {a, c}. 

Since a ∈ A and a ∉ B.Then, A ⊄ B 

Now, b ∈ B and b ∉ C ⇒ B ⊄C. 

But, A ⊂ C since, a ∈ A and a ∈ C. 

(vi) False. 

Explanation: Let A = {x}, B = {{x}, y} 

Now, x ϵ A and {x} is an element of B ⇒ A ϵ B 

But, x is not an element of B. Thus, x∉B.