Math 105A Homework 1

Please complete the following from the textbook:

  • Section 1.1 - #1
  • Section 1.2 - #1-4
  • Section 1.3 - #1,2
  • Section 1.4 - #1

Transcribed Questions

1.1.1

Show that the number $\sqrt[3]{2} + \sqrt[3]{3}$ is algebraic

1.2.1

Show that $f: N \times N \rightarrow N$ defined by $f(m,n) = 2^{m-1}(2n-1)$ is a bijective function

1.2.2

Create an injective map $f: \mathbb{N}^k \rightarrow \mathbb{N}$ and show that your creation is injective

1.2.3

Show that the algebraic numbers are countable by making use of the following:

  • For a polynomial of degree $n$ with integer coefficients, $p(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$, explain without proof the cardinality of the set ${a | p(a) = 0}$.
  • Create a surjection from $\mathbb{Z}^{n+1} \times {1, 2, \dots, n}$ to zeroes of any nth degree integer coefficient polynomial. Explain why $\mathbb{Z}^{n+1}\times{1, 2, \dots , n}$ is countable by quoting theorems in this section.
  • Explain why the set of algebraic numbers is countable using the prior part and quoting a theorem in this section.

1.2.4

Show that if a countable subset is removed from an uncountable set, the remainder is still uncountable

1.3.1

Prove that if $a > 0$ and $b > 0$ then $a + b > 0$

1.3.2

Prove proposition 129. The axioms for multiplication imply the following:

  • If $x \neq 0$ and $xy = xz$, then $y = z$
  • If $x \neq 0$ and $xy = x$, then $y = 1$
  • If $x \neq 0$ and $xy = 1$, then $y = \frac1x$
  • If $x \neq 0$, then $x = \frac{1}{\frac1x}$

1.4.1

Prove $||x| - |y|| \leq |x - y|$