12 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

2.2 Exercises1. Consider the expression x+ y(x+ y)− x(y− x)≡ f (x,y) . Find f (−1,2) .

2. Show −(ab) = (−a)b.

3. Show on the number line the effect of multiplying a number by −1.

4. Add the fractions xx2−1 +

x−1x+1 .

5. Find a formula for (x+ y)2 ,(x+ y)3 , and (x+ y)4 . Based on what you observe forthese, give a formula for (x+ y)8 .

6. When is it true that (x+ y)n = xn + yn?

7. Find the error in the following argument. Let x= y= 1. Then xy= y2 and so xy−x2 =y2− x2. Therefore, x(y− x) = (y− x)(y+ x) . Dividing both sides by (y− x) yieldsx = x+ y. Now substituting in what these variables equal yields 1 = 1+1.

8. Find the error in the following argument.√

x2 +1 = x + 1 and so letting x = 2,√5 = 3. Therefore, 5 = 9.

9. Find the error in the following. Let x = 1 and y= 2. Then 13 = 1

x+y =1x +

1y = 1+ 1

2 =32 . Then cross multiplying, yields 2 = 9.

10. Find the error in the following argument. Let x = 3 and y = 1. Then 1 = 3− 2 =3− (3−1) = x− y(x− y) = (x− y)(x− y) = 22 = 4.

11. Find the error in the following. xy+yx = y+y = 2y. Now let x = 2 and y = 2 to obtain

3 = 4.

12. Show the rational numbers satisfy the field axioms. You may assume the associative,commutative, and distributive laws hold for the integers.

13. Show that for n a positive integer, ∑nk=0 (a+bk) = ∑

nk=0 (a+b(n− k)) . Explain why

2n

∑k=0

(a+bk) =n

∑k=0

2a+bn = (n+1)(2a+bn)

and so ∑nk=0 (a+bk) = (n+1) a+(a+bn)

2 .

2.3 Set NotationA set is just a collection of things called elements. Often these are also referred to as pointsin calculus. For example {1,2,3,8} would be a set consisting of the elements 1,2,3, and8. To indicate that 3 is an element of {1,2,3,8} , it is customary to write 3 ∈ {1,2,3,8} .9 /∈ {1,2,3,8} means 9 is not an element of {1,2,3,8} . Sometimes a rule specifies a set.For example you could specify a set as all integers larger than 2. This would be written asS = {x ∈ Z : x > 2} . This notation says: the set of all integers, x, such that x > 2.

If A and B are sets with the property that every element of A is an element of B, thenA is a subset of B. For example, {1,2,3,8} is a subset of {1,2,3,4,5,8} , in symbols,