Read: Section 2.3.
Turn in: to 2.46, 2.47, 2.50, 2.51, 2.52
- There are many possible answers for 2.46 (so try to be creative), but if you’re stuck, try to modify the proposition from 2.45 to always be true instead of always false.
- Remember, for 2.51 and 2.52, the books asks you to prove the theorem by proving the contrapositive. Make sure to go slowly and carefully when writing out the contrapositive, but do note that it can be done different ways.
- For example, 2.51 may be rewritten as: “Let $n\in \mathbb{Z}$. If $n^2$ is even, then $n$ is even.” The first sentence may be viewed as what the book calls an upfront assumption, and the second sentence is an instance of $A \implies B$. So the contrapositive is $\neg B \implies \neg A$, or in words: “if $n$ is odd, then $n^2$ is odd.” Alternatively, 2.51 could be rewritten as: “If $n\in \mathbb{Z}$ and $n^2$ is even, then $n$ is an even integer.” This version is now an instance of $(A \wedge C) \implies B$. (But I personally favor the first version that treats $n\in \mathbb{Z}$ as an upfront assumption.)
Extra practice: 2.45