Math 104 Assignment 1 common mistakes

• In Problem 1:
  • Assuming $\sup(E) \in E$.
  • Assuming $E \subseteq \R$ rather than an arbitrary ordered set $S$ (for $x, y \in S$, there may be no meaningful notion of $x+y$).

  • Not a mistake, per se, but adding an unnecessary assumption towards a contradiction.

    Consider the following argument that any number divisible by $4$ is divisible by $2$:

    Towards a contradiction, assume that $n \in \N$ is divisible by $4$ but not divisible by $2$. Since $n$ is divisible by $4$, we may write $n = 4k$ for some $k \in \N$. But then $n = 2(2k)$ and $2k \in \N$, so $n$ is divisible by $2$. This contradicts our original assumption, and so the claim is proved.
    It is cleaner to argue directly:
    Suppose $n$ is divisible by $4$. We may therefore write $n = 4k$ for some $k \in \N$. But then $n = 2(2k)$ and $2k \in \N$, so $n$ is divisible by $2$, and the claim is proved.

    There is a similar temptation when arguing the contrapositive. Consider the following argument that if $x^2$ is even, then $x$ is even:

    Towards a contradiction, suppose that $x^2$ is even while $x$ is odd. The product of two odd numbers is odd, so $x\cdot x = x^2$ is odd. This contradicts our assumption that $x^2$ is even, and the claim is established.
    Again, it's cleaner to just argue the contrapositive itself:
    To establish the claim, we will argue its contrapositive. Suppose therefore that $x$ is odd. The product of two odd numbers is odd, so $x\cdot x = x^2$ is odd, and the claim holds.

• In general, but especially in 4.A., there was a tendency to give intuitive arguments rather than actual proofs.
• For 5.C., assuming that the product or sum of irrational numbers is irrational.