1 Logic / 3 pts
(a) Using a truth table, show that \(\lnot(p\Rightarrow q)\) and \(p\land\lnot q\) are equivalent. (b) Write the negation of “For every real number \(x\) there is an integer \(n\) with \(n>x\).”
2 A proof / 3 pts
Prove that if \(n\) is an integer and \(n^2\) is odd, then \(n\) is odd. (Hint: use the contrapositive.)
3 Induction / 4 pts
Prove by induction that \(1^2+2^2+\cdots+n^2=\dfrac{n(n+1)(2n+1)}{6}\) for all \(n\ge1\).
4 Sets and functions / 3 pts
(a) How many integers from 1 to 12 are divisible by 2 or by 3? (b) How many functions are there from \(\{1,2,3\}\) to \(\{a,b,c,d\}\)? (c) How many of them are injective?
5 Counting and graphs / 4 pts
A club has 9 members. (a) In how many ways can it choose a president, a vice president and a treasurer? (b) In how many ways can it choose a 4-person committee? (c) How many edges does the complete graph \(K_6\) have? (d) A graph has 6 vertices with degrees 3, 3, 3, 3, 2, 2. How many edges does it have?
6 A recurrence / 3 pts
Let \(a_0=1\) and \(a_n=2a_{n-1}+3\). (a) Compute \(a_1,a_2,a_3\). (b) Prove by induction that \(a_n=2^{n+2}-3\).
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