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)

It is given that if x∈A and A∈B, then x∈B.

The given statement is false.

Consider that A={ a,b } and B={ a,{ a,b },c }.

Here, b∈{ a,b } and { a,b }∈{ a,{ a,b },c }.

So, it is clear that A∈B.

But, b∉{ a,{ a,b },c }.

(ii)

It is given that if A⊂B and B∈C, then A∈C.

The given statement is false.

Consider that A={ a }, B={ a,b } and C={ c,{ a,b },d }.

It is clear that A⊂B and B∈C, but since a∉{ c,{ a,b },d }, so, A∉C.

(iii)

It is given that if A⊂B and B⊂C, then A⊂C.

The given statement is true.

Consider that x∈A.

Since A⊂B and B⊂C, then,

x∈B and x∈C

Therefore, A⊂C.

(iv)

It is given that if A⊄B, and B⊄C, then A⊄C.

The given statement is false.

Consider that A={ 0,1 }, B={ 2,3,4 } and C={ 1,0,2,3,5 }

It is clear that A⊄B, and B⊄C, but A⊂C.

(v)

It is given that if x∈A and A⊄B, then, x∈B.

The given statement is false.

Consider that A={ a,b,c } and B={ a,d,e }

Here b∈A and A⊄B, but b∉B.

(vi)

It is given that if A⊂B and x∉B, then x∉A.

The given statement is true.

Let x∈A, then as it is given that A⊂B, so, x∈B which is a contradiction to the given condition that x∉B.

Therefore, x∉A.