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
Answers
Given : (i) x∈A and A∈B, then x∈B (ii) If A⊂B and B∈C, then A∈C
To find : Show which are true & Which are False
Solution:
⊂ - Subset
A ⊂ B ( all elements of A are in B & B may have extra elements)
∈ - Belongs to elements in set
(i) If x∈A and A∈B, then x∈B
FALSE
Let say A = { 1 , 2 } & B = { {1 , 2} , 3 } => A∈B
x = 1 , x = 2 as x ∈A
1 , 2 ∉ { {1 , 2} , 3 }
=> x ∉B
(ii) If A⊂B and B∈C, then A∈C
FALSE
A = { 2} B = { 1 , 2 } C = { {1 , 2} ,3 }
A⊂B B∈C
{ 2} ∉ C = { {1 , 2} ,3 }
=> A ∉C
(iii) If A⊂B and B⊂C, then A⊂C
TRUE
A = { 1 , 2} B = { 1 , 2 , 3 } C = { 1 , 2 , 3 , 4 }
A⊂B and B⊂C
{ 1 , 2} ⊂ { 1 , 2 , 3 , 4 }
=> A⊂C
(iv) If A⊄B and B⊄C, then A⊄C
FALSE
A = { 1 , 2 } B = { 3 , 4 } C = { 1 , 2 }
A⊄B , B⊄C, but { 1 , 2 } ⊂ { 1 , 2 }
=> A ⊂C
(v) If x∈A and A⊄B, then x∈B
FALSE
A = { 1 , 2 } B = { 3 }
x = 1 , 2
1 , 2 ∉ { 3 }
=> x ∉ B
(vi) If A⊂B and x∉B, then x∉A
TRUE
A = { 1 } B = { 1 , 2}
x = 3
3 ∉B
3 ∉ { 1 }
=> x ∉ A
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