# Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 2

## Contents

# Question 2.1.1[edit]

## 1[edit]

If is a real number, then the area of a circle of radius is .

## 2[edit]

If there is a line and a point not on , then there is exactly one line containing that is parallel to .

## 3[edit]

If is a triangle with sides of length and then

## 4[edit]

If is a continuous function on [a, b] and is an function such that , then...

# Question 2.2.2[edit]

## 1[edit]

If , then there is an integer q such that . Let q = n.

## 2[edit]

If , then there is an integer q such that . Let q = 1.

## 3[edit]

If , then there is an integer q such that . This implies , and so , and thus .

# Question 2.2.3[edit]

## 1[edit]

If n is an even integer, then for some integer k, .

Let .

Then .

## 2[edit]

If n is an odd integer, then for some integer k, .

Let .

Then .

## 3[edit]

If n is even, then . For integers j and k, let .

, so is even.

If n is odd, then . For integers j and k, let .

, so is odd.

# Question 2.2.6[edit]

If a|b, and b|bm then a|bm, implying aj = bm for some integer j.

Also, if a|c, and c|cn then a|cn, implying ai = cn for some integer i.

We let x = (j+i).

ax = aj+ai

ax = bm+cn

Which implies a|(bm+cn).

# Question 2.2.7[edit]

implies that for some integer, x.

implies that for some integer, y.

Therefore,

for some integer, j.

Let , hence .

# Question 2.3.3[edit]

Suppose that . This means there is an integer such that . Then, we have:

We may consider the integer . Therefore, we have that . Then