InterviewSolution
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InterviewSolution
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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`. |
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Answer» (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`. |
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