In: Math
Problem 2.3 Suppose that we take the universal set U to be the integers. Let S be the even integers, letT be the integers that can be obtained by tripling any one integer and adding one to it, and let V be the set of numbers that are whole multiples of both two and three.
(i) Write S, T, and V using symbolic notation.
(ii) ComputeS∩T, S∩V andT∩V and give symbolic representations that do not use the symbols S, T, or V on the right-hand side of the equals sign.
Since, U is the set of integers
=> U = {...,-2,-1,0,1,2,3,...}
(i)
Since, S is the set of even integers
=> S = {...,-4,-2,0,2,4,...}
Thus,
Now, T is the set of integers that can be obtained by tripling any one integer and adding one to it
Thus, T =
Also, V is the set of numbers that are whole multiples of both two and three. Thus, V is the set of multiples of 6. And thus:
(ii)
Thus,
For , we observe that:'
If a number say y belongs to V, then it is a multiple of 6, thus, y is also divisible by 2 and thus y also belongs to S.
Since, y is arbitrary, it is true for all elements of V and hence
Thus,
For , we observe that:
All the elements of V are divisible by 6 and thus are also divisible by 3.
None of the elements of T are divisible by 3 since they are of the form "3x+1"
Thus,
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