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 \(x ∉ B \)
\(∴ x ∉ A\)