Exam 14: Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

(a) Let $m$ be a positive integer greater than 2 . Show that the relation $R$ consisting of those ordered pairs of integers (a, b) with $a \equiv \pm b$ (mod m) is an equivalence relation. (b) Describe the equivalence classes of this relation where $m$ =4 .

Find a recurrence relation and initial conditions for the number of ways to go up a flight of stairs if stairs can be climbed one, two, or three at a time.

How many nonisomorphic unrooted trees are there with four vertices? Draw these trees.

A fair coin is flipped until a tail first appears, at which time no more flips are made. (a) What is the probability that exactly five flips are made? (b) What is the expected number of flips?

Find a spanning tree for the graph $K _ { 3,4 }$ using (a) a depth-first search. (b) a breadth-first search.

Prove or disprove that the fourth power of an odd positive integer always leaves a remainder of 1 when divided by 16.

(a) Prove or disprove: If a ≡ b (mod 5), where a and b are integers, then $a ^ { 2 } \equiv b ^ { 2 } ( \bmod 5 )$ (b) Prove or disprove: If $a ^ { 2 } \equiv b ^ { 2 } ( \bmod 5 )$ where a and b are integers, then a ≡ b (mod 5).

(a) Show that the relation $R = \{ ( x , y ) \mid x - y \text { is an even integer } \}$ is an equivalence relation on the set of real numbers. (b) What are the equivalence classes of $1 \text { and } \frac { 1 } { 2 } \text { with respect to } R \text { ? }$

Prove or disprove that $\overline { ( A - B ) } = \bar { A } \cup B \text { whenever } A \text { and } B \text { are sets. }$