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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

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Solution

(i)

It is given that if xA and AB, then xB.

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 AB.

But, b{ a,{ a,b },c }.

(ii)

It is given that if AB and BC, then AC.

The given statement is false.

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

It is clear that AB and BC, but since a{ c,{ a,b },d }, so, AC.

(iii)

It is given that if AB and BC, then AC.

The given statement is true.

Consider that xA.

Since AB and BC, then,

xB and xC

Therefore, AC.

(iv)

It is given that if AB, and BC, then AC.

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 AB, and BC, but AC.

(v)

It is given that if xA and AB, then, xB.

The given statement is false.

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

Here bA and AB, but bB.

(vi)

It is given that if AB and xB, then xA.

The given statement is true.

Let xA, then as it is given that AB, so, xB which is a contradiction to the given condition that xB.

Therefore, xA.


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