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

(i) Let us assume A = {1, 2}


And, B = {1, {1, 2}, {3}}


So,


And,


∴


But,


Hence, the given statement is false


(ii) Let us assume, A {2}


B = {0, 2}


And, C = {1, {0, 2}, 3}


It is given in the question that,



∴


But,


Hence, the given statement is false


(iii) It is given in the question that,



Let us assume,


So,


And,


∴


Hence, the given statement is correct


(iv) It is given in the question that,


A ⊄ B


And, B ⊄ C


Let us now assume, A = {1, 2}


B = {0, 6, 8}


And, C = {0, 1, 2, 6, 9}


∴


Hence, the given statement is false


(v) It is given in the question that,



And, A ⊄ B


Let us now assume, A = {3, 5, 7}


And, B = {3, 4, 6}


As,


And,


∴


Hence, the given statement is false


(vi) It is given in the question that,


A ⊂ B


And, x ∉ B


Let us suppose, then we have:



But it is given in the question that, x ∉ B


∴ x ∉ A


Hence, the given statement is true


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