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

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 = {2, 3} and B = {6, {2, 3}, 8}
Here 2 and 3 belongs to A and A belongs to B but 2 and 3 are not belongs to C. Hence statement is false.
(ii) False
Let A ={1,3} B = {1 , 3, 5} and C ={{1 , 3, 5}}
In above example
A ⊂ B and B ∈ C but A ∉ C. Hence statement is false.

(iii) True
Let A ⊂ B and B ⊂ C.
Let x ∈ A it is given that A ⊂ B, so that
x ∈ B it is given that B ⊂ C, so that
x ∈ C
Hence A ⊂ C .
(iv) False
Let A ={1, 2, 3} , B = {2, 3, 4} and C = {1, 2, 3, 5}
In above example
A ⊄ B and B ⊄ C, but A ⊂ C
Hence given statement is false.
(v) False
Let A = {1 , 2 , 3}, B = {2, 3, 4}
In above example if x = 1
1 ∈ A and A ⊄ B but 1 ∉ B. Hence given statement is false.
(vi) True
Let x ∉ B and A ⊂ B so that all element of set A will belongs to set B. if set B don’t have element x then A cannot have this element.