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 $\in$ A and A $\in$ B, then x $\in$ B
(ii) If A $⊂$ B and B $\in$ C, then A $\in$ 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
Let A = {1, 2} and B = {1, {1, 2}, {3}}
Now,
2 $∈$ {1,2} and {1,2} $∈$ {{3},1,{1,2}}
$∴ A ∈ B$
However,
2$∉$ {{3},1,{1,2}}

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(ii) False
Let
A = {2}, B= {0,2}, and C = {1{0,2},3}
As $A ⊂ B$
$B ∈ C$
However, $A ∉ C$

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(iii) True
Let $A ⊂ B$ and $B ⊂ C$.
Let $x ∈ A$
$⇒ x ∈ B [∴ A ⊂ B]$
$⇒ x ∈ C [∴ B ⊂ C]$
$∴ A ⊂ C$

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(iv) False
A = {1,2}, B = {0,6,8}, and C = {0,1,2,6,9}
$A ⊄ B$ and $B ⊄ C$
Let Accordingly,
However, $A ⊂ C$

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(v) False
Let A = {3, 5, 7} and B = {3, 4, 6}
Now, $5 ∈ A$ and $A ⊄ B$
However, $5 ∉ B$

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(vi) True
Let $A ⊂ B$ and $x ∉ B$.
To show: $x ∉ A$ If possible,
suppose $x ∈ A.$
Then, $x ∈ B$, which is a contradiction as $x ∉ B$
$∴ x ∉ A$