• Turn in February 11th: Write down both the converse and contrapositive of the statement:

If $$x^2 = x + 1$$, then $$x = 1 + \sqrt{5}$$ or $$x = 1 - \sqrt{5}$$

• Turn in February 13th:

Prove the following: If $$x^3$$ is irrational then $$x$$ is irrational.

• Turn in February 16th:

Let $$n = ab$$ be the product of positive integers $$a$$ and $$b$$. Prove that either $$a \leq \sqrt{n}$$ or $$b \leq \sqrt{n}$$.

• Turn in February 18th:

Show that $$(p \to q) \to r$$ is not logically equivalent to $$p \to (q \to r)$$

• Turn in February 20th:

Simplify the following if possible: $$(p \to (q \to r)) \to ((p \to q) \to (p \to r))$$

• Turn in February 23rd:

Let $$a$$ be an integer. Prove that there exists an integer $$k$$ such that $$a^2 = 3k$$ or $$a^2 = 3k + 1$$.

• Turn in February 25th:

Let $$x,a$$ be integers with $$a \geq 2$$ such that $$a \mid 11x + 3$$ and $$a \mid 55x + 52$$. Find $$a$$.

• Turn in February 27th:

Tell me how many natural numbers $$1 \leq k \leq 14$$ smaller than 15 have the following property $$\gcd{(k, 15)} = 1$$. Just for your information, we say two numbers $$a, b$$ are relatively prime when $$\gcd{(a,b)} = 1$$. Also (for your information only) the number of relatively prime natural numbers smaller than $$a$$ is called $$\phi(a)$$ the Euler Totient Function. So this question is asking you to calculate $$\phi(15)$$.

• Turn in March 2nd:

Determine whether 9833 is prime.

• Turn in March 4th:

Show that for every natural number $$k$$, $$6^k$$ will never end in a 0 (using a decimal representation).

• Turn in March 13th:

Prove that $$6 | n^3 + 5n$$ for all $$n \geq 1$$

• Turn in March 16th:

Find a (closed-form) formula for the following sum: $$(0 - 0) + (1 - 1) + (8-4) + (27-9) + \cdots + (n^3 - n^2)$$ Use induction to prove your result.

• Turn in March 18th:

Find this sum: $$S = -56 - 49 - 42 - \cdots + 623 + 630 + 637$$

• Turn in March 20th:

Which, if any, of the following three sets are equal? $$A = \{ k \mid k \in \mathbb{N}, 3 \nmid k, k \leq 12 \}$$ $$B = \{ 3k + 1 \mid k \in \mathbb{N}, k \leq 4 \}$$ $$C = \{ k \mid k \in \mathbb{N}, \gcd{(k, 3)}=1, k \leq 11 \}$$

• No Problem for March 23rd
• Turn in March 25th

Given the set $$A = \{1, 2, 3\}$$ create a binary relation on $$A$$ (that is, a subset of $$A \times A$$) which is reflexive,symmetric, and is not transitive

• Turn in March 27th

Prove that $$\overline{a} + \overline{b} = \overline{a+b}$$ where $$\overline{a}$$ is the equivalence class of $$a$$ under the relation on $$\mathbb{Z}$$ given by $$x \sim y$$ when $$p | (x-y)$$. Here $$\overline{a} + \overline{b}$$ means the set $$\{ x + y \mid x \sim a \land y \sim b \}$$, made by summing anything in $$\overline{a}$$ and anything in $$\overline{b}$$. This is an elaborate way of asking if computing $$(a + b) \textrm{ mod } p = (a \textrm{ mod } p) + (b \textrm{ mod } p)$$.

• Turn in April 6th

None, happy spring break.

• Turn in April 8th

Let $$S = \{ 1,2,3,4\}$$ and $$f,g : S \to S$$ by $$f = \{(1,3),(2,2),(3,4),(4,1)\}$$ and $$g = \{(1,4), (2,3), (3,1), (4,2)\}$$. Find $$g \circ f \circ g^{-1}$$.

• Turn in April 13th

Find $$x$$ such that $$79x \equiv 15 \mod{ 722 }$$

• Turn in April 15th

Find $$6^{128} \mod{ 13 }$$

• None on April 17th
• Turn in April 20th

Problem 5 from math.prof.ninja/417, the in-class problems from Friday.

• Turn in April 22th

Problem 8 from math.prof.ninja/420, the in-class problems from Monday.

• Turn in April 24th

Problem 6 from math.prof.ninja/422

• Turn in April 27th

The last problem from 424

• Turn in April 29th

Give a method for detecting if a graph is bipartite.

• Turn in May 1st

How many 6 digit integers have 1 digit repeated 3 times, another distinct digit repeated twice, and a third digit? For instance 292129 or 877666.

• Turn in May 4th

Prove that the Petersen graph is not Hamiltonian

• Turn in May 6th

Create a graph in which any walk on the graph of length $$k$$ is a length $$k$$ word made from 'a' and 'b' with at most two 'b's in a row. Use this graph to decide the number of words of length 4 with this property.

• May 8th Test 3, good luck.
• May 11th, new topic no daily challenge.
• May 13th:

Problem 4 from Monday's problem set

• May 15th:

What is the worst case complexity of insertion sort?

• May 18th, final class:

Make a list of the super-powers you developed (or are developing) in this class.