MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2011
Solutions to Selected Problems from Problem Set C
1. 1.17. The domain of y = jxj is R (every number has an absolute value).
The image is [0;1) since every y ▯ 0 is the absolute value of itself: y = jyj .
1.35. x=y + y=x ▯ 2 if and only if x and y have the same sign: If x or y is 0, then the expression
is unde▯ned. If they have opposite signs, then the left side is negative. If they have the same
2 2 2 2
sign, then multiplying by xy yields x +y ▯ 2xy, equivalent to x ▯2xy+y ▯ 0, equivalent to
(x ▯ y) ▯ 0. The last inequality holds whenever x and y have the same sign, so this necessary
condition is also su▯cient.
1.49. (a) TRUE. If f and g are bounded, then there exist M an1 M , suc2 that jf(x)j ▯ M 1 and
jg(x)j ▯ M 2 for all x 2 R . Using the triangle inequality, we conclude tha