For any two sets A and B, show that the following statements are equiv

1.	For any two sets A and B, show that the following statements are equivalent:(i) A ⊂ B(ii) A – B = ϕ(iii) A ∪ B = B(iv) A ∩ B = A
Answer» (i) A ⊂ B To prove that the following four statements are equivalent, we need to prove (i)=(ii), (ii)=(iii), (iii)=(iv), (iv)=(v) Now, firstly let us prove (i)=(ii) As we know, A–B = {x ∈ A: x ∉ B} as A ⊂ B, Therefore, each element of A is an element of B, ∴ A–B = ϕ Thus, (i)=(ii) (ii) A – B = ϕ As we need to show that (ii)=(iii) On assuming A–B = ϕ To prove: A∪B = B ∴ Every element of A is an element of B Therefore, A ⊂ B and so A∪B = B Thus, (ii)=(iii) (iii) A ∪ B = B As we need to show that (iii)=(iv) On assuming A ∪ B = B To prove: A ∩ B = A. ∴ A⊂ B and so A ∩ B = A Thus, (iii)=(iv) (iv) A ∩ B = A Finally, we need to show (iv)=(i) On assuming A ∩ B = A To prove: A ⊂ B Thus, A ∩ B = A, therefore A ⊂ B Thus, (iv) = (i)

For any two sets A and B, show that the following statements are equivalent:(i) A ⊂ B(ii) A – B = ϕ(iii) A ∪ B = B(iv) A ∩ B = A

Answer»

(i) A ⊂ B

To prove that the following four statements are equivalent, we need to prove (i)=(ii), (ii)=(iii), (iii)=(iv), (iv)=(v)

Now, firstly let us prove (i)=(ii)

As we know, A–B = {x ∈ A: x ∉ B} as A ⊂ B,

Therefore, each element of A is an element of B,

∴ A–B = ϕ

Thus, (i)=(ii)

(ii) A – B = ϕ

As we need to show that (ii)=(iii)

On assuming A–B = ϕ

To prove: A∪B = B

∴ Every element of A is an element of B

Therefore, A ⊂ B and so A∪B = B

Thus, (ii)=(iii)

(iii) A ∪ B = B

As we need to show that (iii)=(iv)

On assuming A ∪ B = B

To prove: A ∩ B = A.

∴ A⊂ B and so A ∩ B = A

Thus, (iii)=(iv)

(iv) A ∩ B = A

Finally, we need to show (iv)=(i)

On assuming A ∩ B = A

To prove: A ⊂ B

Thus, A ∩ B = A, therefore A ⊂ B

Thus, (iv) = (i)