Show:
\(P \rightarrow Q\)
4-Letter Words
Potent Potables
Sequences and Sums
True or False
100
\(Q\rightarrow P\)
What is the converse?
100
The number of 4-letter words.
What is \(26^4\), or 456976?
100
It's the proof technique you would use to show that a graph is not planar.
What is a proof by contradiction?
100
1, 3, 6, 10, 15, 21, 28, ...
What are the Triangular numbers?
100
If a graph has chromatic number 5, then no matter how you draw it, there will be edges crossing.
What is True? This is the contrapositive of the 4-color theorem.
200
\(\neg Q \rightarrow \neg P\)
What is the contrapositive?
200
The number of 4-letter words containing no repeated letters.
What is \(P(26,4)\), or 358800?
200
To prove that \(n\binom{n-1}{k-1} = k\binom{n}{k}\) you could give an algebraic proof or this kind of proof.
What is a combinatorial proof?
200
The recursive definition of the sequence \(2, 5, 7, 12, 19, 31,\ldots\)
What is \(a_n = a_{n-1} + a_{n-2}\); \(a_0 = 2\) and \(a_1 = 5\)?
200

\(\neg(P\rightarrow Q) \leftrightarrow (\neg P \rightarrow\neg Q)\)

What is False?

300
\(P \wedge \neg Q\)
What is the negation?
300
The number of 4-letter words in which the letters are in alphabetical order.
What is \({29 \choose 4}\), or 23751?
300
This is what you would use to prove a fact about a recurrence relation.
What is proof by mathematical induction?
300
A closed formula for \(1+3+5+7+\cdots + (2n+1)\)
What is \(n^2\)?
300
\((P \rightarrow Q) \vee P\)
What is True?
400

The negation of the converse of the contrapositive

What is \(\neg P \wedge Q\) or \(\neg (\neg P\rightarrow \neg Q)\)?

400
The number of 4-letter words in which the letters are in alphabetical order with no repeats.
What is \({26 \choose 4}\), or 14950?
400

A style of "proof" that is never sufficient to prove a universal statement

What is proof by example?

400
\[\sum_{k = 0}^{10} {10 \choose k}\]
What is \(2^{10}\)?
400
If \(|A \cup B| = 5\) then \(|A \cap B| \ne 5\) or \(|A| = 5\)
What is True?
500

An equivalent disjunction.

What is \(\neg P \vee Q\)?

500
The number of 4-letter words which use all of the letters in "for'' (and no others).
What is \(36\), or \({4 \choose 2}\cdot 3!\)?
500
Used to establish an implication \(P \rightarrow Q\) by first assuming \(\neg Q\).
What is a proof by contrapositive?
500
A binomial coefficient equal to the sum of the first 10 triangular numbers.
What is \({12 \choose 3}\)
500
If a graph contains exactly 3 vertices with odd degree, then the graph contains an Euler circuit.
What is True? (The hypothesis is false for all graphs.)