Let G be the graph with vertices v_{1}, v_{2} and v_{3} and the matrix begi

Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols: ${\displaystyle \cap \cup }$

How many elements are in the set { 0, { { 0 } }?

Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.

Let A, B, and C be sets. Show that ${\displaystyle (A-B)-C=(A-C)-(B-C)}$

Prove based on functions
Prove that for all $f,g\in \mathbb{N}\to \{0,1\}$ there exist $h\in \mathbb{N}\to \mathbb{N}$ bijection so:
$f=g\circ h\to f=g$
My attempt was to assume that for all $f,g\in \mathbb{N}\to \{0,1\}$ there exist $h\in \mathbb{N}\to \mathbb{N}$ bijection so: $f=g\circ h\wedge f\ne g$ and try to get a contradiction without any succed.

To determine:
a) Using "Proof by Contraposition", show that: If n is any odd integer and m is any even integer, then, ${\displaystyle 3m^3+2m^2}$ is odd.
b) Using the Mathematical Induction to prove that: ${\displaystyle 3^{2n}-1}$ is divisible by 4, whenever n is a positive integer.

Combinatorial proof of $P_r^{n+1}=r!+r(P_{r-1}^n+P_{r-1}^{n-1}+\cdots +P_{r-1}^r)$, where $P_r^n$ denotes the number of r- permutations of an n element set.