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).