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Integers and Rational Numbers

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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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This solution is comprised of a detailed explanation to answer for any integer a, argue that a + 3 > a + 2.

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Hi,

Please find the solutions/explanations attached herewith.

Solution:

We know that 2 < 3

Then by using the theorem: If a, b, and c are integers and a < b, then a + c < b + c.

2 < 3

Then there exists a such a such that

2 + a < 3 + a

Now by using commutative property of addition of integers, we can say that

a + 2 < a + 3

which means that

a + 3 > a + 2

Thus, for any integer a, a + 3 > a + 2.

Solution:

Given that a is divisible by b then there ...

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Sandhya Sharma, MSc
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  • BSc, Meerut University
  • MSc, Meerut University
  • MPhil, Institute of Advanced Studies
  • MSc, AIT
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