In each of the following, determine whether the statement is true or f

1.	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 BC, 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: For example, let A = {1, 2} and B = (2, 3, {1, 2}}. As 1∈ A and A∈ B but 1∉ B (ii) False: For example, let A = {1, 2}, B = {1, 2, 3} and C = {3, 4,{1, 2, 3}} Now clearly, A ⊂ B and B⊂C but A ∉ C, (iii) True: Given A ⊂ B and Now we have to prove A ⊂ C ∴ x ∈ C ∵ B ∈ C. ∀ X ∈ A ⇒X ∈ C∴ A ∈ C. (iv) False: For example, let A = (1, 2), B = {2, 3),C = (1, 2, 4) Clearly A⊄B and B⊄C but A ⊂ C (v) False: For example, let A=(l,2), B={2, 3, 4, 5} Now 1 ∈ A and A ∉ B but 1 ∉ B (vi) True: Suppose x∈ A then X∈ B A⊂ B.

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 BC, 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: For example, let A = {1, 2} and

B = (2, 3, {1, 2}}. As 1∈ A and A∈ B but 1∉ B

(ii) False: For example, let A = {1, 2}, B = {1, 2, 3} and C = {3, 4,{1, 2, 3}}

Now clearly, A ⊂ B and B⊂C but A ∉ C,

(iii) True: Given A ⊂ B and

Now we have to prove A ⊂ C

∴ x ∈ C ∵ B ∈ C.

∀ X ∈ A ⇒X ∈ C∴ A ∈ C.

(iv) False: For example, let

A = (1, 2), B = {2, 3),C = (1, 2, 4)

Clearly A⊄B and B⊄C but A ⊂ C

(v) False: For example, let A=(l,2), B={2, 3, 4, 5}

Now 1 ∈ A and A ∉ B but 1 ∉ B

(vi) True: Suppose x∈ A then X∈ B A⊂ B.