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 ϵ C. (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.

Question

Philoid · Accepted Answer

(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.