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# Integers in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video. In any case, you will find below a PDF file at the bottom of the step with the solutions. The symbol (in) means “belongs”, so that (xin mathbb R) means that (x) belongs to (mathbb R).

### Exercise 1.

Consider the sets ( (mathbb Z=lbrace …,-3,-2,-1,0,1,2,3,…rbrace))

[ A =leftlbrace 7n^2: nin mathbb{Z},-5leq nleq 5rightrbrace,] [B =leftlbrace dfrac{m(m+1)}2: min mathbb{Z},1leq mleq 5rightrbrace,] [C =leftlbrace 7n^2: nin mathbb{Z}, 0leq nleq 5rightrbrace. ]

1) List all the elements in (A) and in (B).

2) Determine (Acap B) (the intersection of (A) with (B) , (Acup B) (the union of (A) with (B)), (Asetminus C) (the elements that belong to (A) but not to (C)).

### Exercise 2.

Let (A), (B) and (C) be three non-empty subsets of a set (X) such that [Bcap A=Ccap A qquad mbox{and} qquad Bcup A=Ccup A.]

Show that (B = C).