Question

1) Disprove: If r and s are irrational numbers, then rs is irrational

2) Prove that there exist two integers a and b such that a + b > ab.
3) Let m and n be integers. Prove that if m+n≥10,then m≥5 or n≥5

Answer 1

1) Disprove: If r and s are irrational numbers, then rs is irrational.

We can disprove this statement by multiplying two irrational numbers together and getting a rational number. The square root of 2 is an irrational number.

`√(2) *√(2) =2`

2) Prove that there exist two integers a and b such that a + b > ab.

This is asking us to find two numbers that are greater added together than when multiplied. There may be other examples, but we can do this with anything times 1. We can prove this with an example.

7 + 1 > (7)(1) ➜ 8 > 7 ✓

3) Let m and n be integers. Prove that if m + n ≥ 10, then m ≥ 5 or n ≥ 5.

This is saying that if two numbers added together are greater or equal to 10, then one of the numbers must be greater or equal to 5. We can prove this statement by graphing the three inequalities. See attached. You can see that no matter where you look as a solution of m + n ≥ 10, there is an overlap of either m ≥ 5 or n ≥ 5. This means that this statement is true.