HOMEWORK 2
DUE WEDNESDAY JANUARY 21 AT THE BEGINNING OF CLASS
• DNS stands for do not submit
1. Problems
In the following problems, you can use E1-E3, A3 and M3 implicitly. You should pick the
challenges that are appropriate for your level.
(1) Prove one of (a), (b), (c) using the axioms to justify your reasoning. You can consult
Part 2 of John Boller’s notes on my webpage.
(a) (3 + 4) · (3 + 4) = 32 + 2 · 3 · 4 + 42 .
(b) (a + 4) · (a + 4) = a2 + 2 · a · 4 + 42 .
(c) (a + b) · (a + b) = a2 + 2ab + b2 .
OR
Prove one of (i), (ii), (iii) using the axioms to justify your reasoning.
(i) (2 + 4) · (2 + 4) · (2 + 4) = 23 + 3 · 22 · 4 + 3 · 2 · 42 + 43 .
(ii) (a + 4) · (a + 4) · (a + 4) = a3 + 3 · a2 · 4 + 3 · a · 42 + 23 .
(iii) (a + b) · (a + b) · (a + b) = a3 + 3a2 b + 3ab2 + b3 (Hint : Use the fact that
(a + b) · (a + b) = a2 + 2ab + b2 ! )
(2) Read the Rules of Order below and give explicit numerical examples for O1, O3 and
O4.
O1. (Transitivity of Inequality)
If a, b, c ∈ Q and a < b and b < c, the a < c.
O2. (Trichotomy)
If a, b ∈ Q, then exactly one of the following is true: a < b, a = b, or b < a.
O3. (Additive Property of Inequality)
If a, b, c ∈ Q and a < b, then a + c < b + c.
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O4. (Multiplicative Property of Inequality)
If a, b, c ∈ Q, a < b and 0 < c, then a · c < b · c.
2. Extra Practice
(1) DNS Prove that
(a + b + c) · (e + f + g) = ae + af + ag + be + bf + bg + ce + cf + cg.
(2) DNS Think of a general recipe for expanding any products of two sums. By products
of two sums, we mean expressions of the form (a+b+c)·(e+f ), (a+b)·(c+d+a+e),
etc.
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