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.
(i) The statement is false.
Take A = {1}, B = {{1}, 2}
Now 1∈A and A∈B but 1∉B.
(ii) The statement is false
Take A = {1}, B = {1, 2}, C = {{1, 2}, 3}
Now A⊂B and B∈C but A∉C.
(iii) The statement is true.
Let x∈A⇒x∈B (∵ A⊂B)
⇒ x∈C (∵ B⊂C)
Now x∈A ⇒ x∈C
∴ A⊂C.
(iv) The statement is false.
Take A = {1, 2} B = {2, 3 }, C = {1, 2, 5}
Now A⊄B and B⊄C but A⊂C.
(v) The statement is false.
Take A = {1, 2} and B = {2, 3, 4, 5}
Now 1∈A and A⊄B but 1∉B
(vi) The statement is true.
Let x∈A ⇒ x∈B (∵ A⊂B)
Now x∉B ⇒ x∉A.