Prove the following relations by Contradiction (Use rules. Don’t use truth table). a) [B ^^ (B rarr C )] rarr C b) not (p vv not (p ∧ q) ) rarr q

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 } }?

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

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: ${\displaystyle \cap \cup }$

A)If ${\displaystyle X=\{1,2,3,4,\dots 10\},f_A=0101010101\text{ and }B=1,2,4,67,9.\text{ Find }|A\cdot B|}$
B) Let $A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]\text{ and }C=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 1\end{array}\right]$
Find ${\displaystyle (B\cdot C)\cdot A}$

Combinatorics question with unequality and different subscript
a) $x_1+x_2+...+x_7\le 30$ where $x_i^{\prime }s$ are even non-negative integers.
b) $x_1+x_2+...+x_7\le 30$ where $x_i^{\prime }s$ are odd non-negative integers.
c) $x_1+x_2+...+x_6\le 30$ where $x_i^{\prime }s$ are odd non-negative integers.
These questions is from my textbook. I know to solve similar questions such that $x_1+x_2+...+x_k\le n$ where $x_i^{\prime }s$ are non-negative integers. We add an extra term on the lefthandside and the rest found by combination with repetition such that $(\genfrac{}{}{0ex}{}{n+(k+1)-1}{n})$. However , what happens when they are even or odd , is there any special technique ? Moreover , as you see the part a and b differ from only subscripts , i think that there must be a reason behind different subscript in same question.Is there any reason ? Thanks in advance..

Let G be the graph with vertices ${\displaystyle v_1,v_2}$ and ${\displaystyle v_3}$ and the matrix $\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 2& 2& 0\end{array}\right]$
To find the number of walks of from ${\displaystyle v_1}$ to ${\displaystyle v_3}$ we need to find matrix ${\displaystyle A^2}$