# Integers and Rational Numbers

1)For any integer a, argue that a + 3 > a + 2

3) An integer a is divisible by an integer b means there is an integer z such that a = b x z. use any properties of the integers through page 14 to prove that fir integers a,b and c such that if a is divisible by b and b is divisible by c the a is divisible by c.

4) let Z denote the set of integers let S = Z x (Z-{0}). Argue that the relation F defined by F = {((x,y),(u,v)): xv = yu}
is an equivalence on S

5) Consider the previous problem. List five members of the equivalence class (7.4)^F.

6) use the definition of addition of rational numbers to argue that
(3,4)^F + (5,6)^F = (19,12)^F

The attached files are material covered (mathch5) that we can use and
the actual questions are from the questions2.pdf

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#### Solution Preview

...s an integer z such that a = b.z
Since b is divisible by c then there exists an integer x such that b = c.x

Now, a = b.z = (c.x).z = c.(x.z) (by associative property of multiplication of integers)

Since x and z are integers then x.z is also a integer.

Let y = x.z

Then a = c.y

Since there exists a integer y such that a = c.y

Therefore a is divisible by c.

Solution:

Reflexive:
Let (a, b) S. Then (a, b)F(a, b) since ab = ba by commutative property of the ...