NCERT Solutions
Class 11 Maths
Sets
Ex.Misc.Q.2
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.
(1) If x ∈ A and A ∈ B, then x ∈ B
(2) If A ⊂ B and B ∈ C, then A ∈ C
(3) If A ⊂ B and B ⊂ C, then A ⊂ C
(4) If A ⊄ B and B ⊄ C, then A ⊄ C
(5) If x ∈ A and A ⊄ B, then x ∈ B
(6) If A ⊂ B and x ∉ B, then x ∉ A
(1) False.
Let A = {1, 2} and B = {1, {1, 2}, {3}}
Now, 2 ∈ {1, 2} and {1, 2} ∈ {1, {1, 2}, {3}}
Hence, A = B
However, 2 ∉ {1, {1, 2}, {3}}
(2) False.
Let A = {2}, B = {0, 2} and C = {1, {0, 2}, {3}}
As A ⊂ B and B ⊂ C
However, A ∉ C
(3) True.
Let A ⊂ B and B ⊂ C.
Let x ∈ A
=> x ∈ B
[Since A ⊂ B]
=> x ∈ C
[Since B ⊂ C]
So, A ⊂ C
(4) False.
A = {1, 2}, B = {0, 6, 8} and C = {0, 1, 2, 6, 9}
Let A ⊄ B and B ⊄ C
Accordingly,
However, A ⊂ C
(5) False.
Let A = {3, 5, 7} and B = {3, 4, 6}
Now, 5 ∈ A and A ⊄ B
However, 5 ∉ B
(6) True.
Let A ⊂ B and x ∉ B.
To show: x ∉ A
If possible, suppose x ∈ A.
Then, x ∈ B, which is a contradiction as x ∉ B
So, x ∉ A