Question 2:

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

(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, 3}}

(ii) False

Let A = {2}, B = {0, 2} and C = {1, {0, 2}, 3}

As A ⊂ B
B ∈ C

However, A ∉ C

(iii) True

Let A ⊂ B and B ⊂ C.
Let x ∈ A

⇒ x ∈ B      [∵ A ⊂ B]

⇒ x ∈ C         [∵ B ⊂ C]

∴ A ⊂ C

(iv) False

Let, A {1, 2}, B = {0, 6, 8}, and C = {0, 1, 2, 6, 9}

Accordingly, A ⊄ B and B ⊄ C.

However, A ⊂ C

(v) False

Let A = {3, 5, 7} and B = {3, 4, 6}
Now, 5 ∈ A and A ⊄ B

However, 5 ∉ B

(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

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