For the exam on Friday, it would help you to do the following things:
- Learn the definitions you needed to know for the quiz. Know the definitions precisely. You should know the statements in formal language, as well as interpretations in words. E.g., $A\subseteq B$ means $(\forall x)(x\in A \implies x\in B)$ and this can be read as "every element in $A$ is also in $B$".
- "Do blank-paper practice" for the problems on the problem sets that have been handed back, and for the homework problems. Note: You should be able to do all the problems, including the hard ones. Avoid repeating a mistake you made before.
- Understand these challenging concepts:
- $\subseteq$ vs. $\in$ vs. $=$
- $A\subseteq B$ vs. $A\cap B$
- $\P(A)$, the power set of $A$
- $(\forall x)(P(x))$ vs. $\{\,x \mid P(x)\,\}$
- "For all $x$, $\ldots$" vs. "There exists an $x$ where $\ldots$"
- $(\forall x)(\exists y)(P(x,y))$ vs. $(\exists y)(\forall x)(P(x,y))$
- $P \And Q$ vs. $A\cap B$
- $P(x) \implies Q(x)$ vs. $\{\, x \mid P(x) \hbox{ and } Q(x) \,\}$
- Know the negations of the various types of propositions we've studied.
- Know the contrapositive, converse, and inverse, and which ones are equivalent.
- Know how to translate English into formal logic and vice versa.
- Know how to use mathematical induction.
- Know the proofs of these key examples. You should not memorize them, but should remember the central idea(s) and reconstruct the proof from that memorized core.
- $\emptyset\subseteq A$ for all $A$
- $A\subseteq B$ iff $\P(A)\subseteq\P(B)$
- $A\times B=\emptyset$ if, and only if, $A=\emptyset$ or $B=\emptyset$
- For $B\ne\emptyset$, if $A\times B\subseteq C\times B$ then $A\subseteq C$
- $\sqrt{2}$ is irrational
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