Prove that discrete math the following statement (if true) or provide a counterexample (if false): For all n \geq 4, 2^n - 1 is not a prime number.

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: ${\displaystyle \cap \cup }$

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

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

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

Make fractions out of the following information, reduce, if possible,
1 foot is divided into 12 inches. Make a fraction of the distance from 0 to a-d
0 to a. = ___
0 to b. = ___
0 to c. = ___
0 to d. = ___

Partitions of n where every element of the partition is different from 1 is $p(n)-p(n-1)$
I am trying to prove that p(n| every element in the partition is different of $1)=p(n)-p(n-1)$, and I am quite lost... I have tried first giving a biyection between some sets, trying to draw an example in a Ferrers diagram and working on it... Nevertheless, I have not obtained significant results. Then, I have thought about generating functions; we know that the generating function of $\{p(n){\}}_{n\in \mathbb{N}}$ is $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-x^i}$, so $\{p(n-1){\}}_{n\in \mathbb{N}}$ will have $\prod _{i=1}^{\mathrm{\infty }}\frac{x}{1-x^i}$ as generating function. So, what we have to prove is that $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-x^i}-\prod _{i=1}^{\mathrm{\infty }}\frac{x}{1-x^i}=(1-$-x). $\prod _{i=1}^{\mathrm{\infty }}\frac{1}{1-x^i}$ is the generating function of p(n|every element in the partition is different of 1)... but i'm am not seeing why! Any help or hint will be appreciate it!

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