MAT 370/371 Intermediate/Advanced Calculus Aug 25, 2004
Homework for week 1
Problems for section 1.2: Real number axioms, part I - The field axioms
Remember to assume nothing beyond the field axioms and the results that have been proved using
them. Refer to the hand-outs sections on the WWW for the real number axioms.
1. Let x,y, and zbe real numbers. Prove that x(y+z) = xy +xz.
(Note: The textbook has the order reversed – due to a different order in the distributivity axiom.)
3. Prove that −x= (−1) ·xfor every real number x.
5. d. Suppose x, y ∈Rare both nonzero. Prove that
1
x
y
=y
x.
7. If x6= 0 and y6= 0 show that (xy)−1exists and that (xy)−1=x−1y−1.
8. Let A={E, O }be the set containing two elements: The sets of the even and odd integers,
respectively. Define two binary operations on Aas follows:
⊕E O
E E O
O O E
⊗E O
E E E
O E O
Prove that with these operations Asatisfies the field axioms.
Problems for section 1.3: Real number axioms, part II - The order axioms
Use the following Order axiom (i.e. do not use the axioms as stated on the handout on the WWW.)
There exists a subset R+of Rsuch that
•R+is closed under addition and multiplication, i.e. if x, y ∈R+then also x+y∈R+and
xy ∈R+.
•For every a∈Rexactly one of the following holds: a∈R+,a= 0, or a∈R+.
1. Prove that if a∈Rand b∈R+then a < a +b.
3. Let a, b ∈R. Prove that if a < 0 and b < 0 then a+b < 0 and a·b > 0.
5. Prove that 1
2<1.
7. Prove the law of trichotomy: For every pair of real numbers aand bexactly one of the following is
true: a < b or a=bor a > b.
9. b. Suppose a∈Rand 0 < a < 1. Prove a2< a.
9. d. Suppose a, b ∈Rhave the same sign. Prove that if a≤bthen 1
a≥1
b.
11. Prove that there is no rational number whose square is 2.
Hint: Proceed by contradiction. Assume that there is some rational number m
nwhose square is 2
and derive a contradiction.
