Tuesday, April 15, 2014

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.


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