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For the exam on Friday, it would help you to do the following things:
- Learn the definitions you needed to know for the quiz, plus the definition of a function. Know the definitions precisely. You should know the statements in formal language, as well as interpretations in words. E.g., "
AÍB " means "("x)(xÎA Þ xÎB) " and this can be read as "everything in A is also in B".
- Do "blank-paper practice" for the problems on the four problem sets that have been handed back. Note: You should be able to do all the problems, including the hard ones. Avoid repeating a mistake you made on the problem set.
- Understand these challenging concepts:
- Í vs. Î vs. =
- AÍB vs. AÇB
- P(A), the power set of A.
- ("x)(P(x)) vs.
{ x | P(x) } - "For all x, ..." vs. "There exists an x where ..."
- ("x)($y)(P(x,y)) vs. ($y)("x)(P(x,y))
- PÙQ vs. AÇB
P(x) Þ Q(x) vs.{ x | P(x) 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.
- ØÍA for all A
(theorem 9/26/01) - AÍB if, and only if, P(A)ÍP(B)
(theorem 10/1/01) - A´B=Ø if, and only if, A=Ø or B=Ø (PS4#4)
- For A¹Ø and B¹Ø, A´BÍC´D iff AÍC and BÍD (PS4#6)
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