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 in A and A in B`, then `x in B`
(ii) If `A sub B and B in C`, then `A in C`
(iii) If `A sub B and B sub C`, then `A sub C`
(iv) If `A cancel(sub)B and B cancel(sub) C`, then `A cancel(sub)C`
(v) If `x in A and A cancel(sub)B`, then `x in B`
(vi) If `A sub B and x cancel(in) B`, then `x cancel(in)A`.

Explanation

(i) Statement, if `x in A and A in B` then `x in B` is false
Example : Let `A = {2} and B = {{2},3}`
It is clear that `2 in A and A in B` but `A cancel(in)B`
(ii) Statement, if `A sub B and B in C` then `A in C` is false.
Example : Let `A = {1},B={1,2} and C={1,2},3,4}`
It is clear that `A sub B and B in C` but `A cancel(in)C`
(iii) Statement, if `A sub B and B sub C`, then `A sub C` is true
Let `x in A`
`:. A sub B and x in A` then `x in B`
but `B sub C and x in B` then `x in C`
`because x in A rArr x in C`
`:. A sub C`
Therefore, if `A sub B and B sub C`, then `A sub C`
(iv) Statement, if `A cancel(sub)B and B cancel(sub)C`, then `A sub C` is false
Example : Let `A = {1,2}, B = {3,4} and C= {1,5}` then elements 1 and 2 of A are not in B from which `A cancel(sub)B` and elements 3 and 4 of B are not in C from which `B cancel(sub)C`
But element 2 of A is not in C from which `A cancel(sub)C`
(v) Statement, if `x in A and A cancel(sub)B`, then `x in B` is false
Example : Let `A = {1,2} and B = {2,3,4}`
It is clear that `1 in A and A cancel(sub)B` then `1 cancel(in)B`
(vi) Statement, if `A sub B and x cancel(in)B` then `x cancel(in)A` is true i.e, all elements of A are also in B
i.e., if any element is not in B then it will not also be in A
i.e., `x cancel(in)B rArr x cancel(in)A`
Therefore, if `A sub B and x cancel (in) B` then `x cancel(in) A`
Therefore, if `A sub B and x cancel (in) B` then `x cancel (in) A`.