Explore Discrete Math Examples

${\displaystyle \text{Let}U=\{p,q,r,s,t\},D=\{p,r,s,t\},E=\{q,s\},F=\{p,r\},\text{and}G=\left\{s\right\}}$
Determine whether the statement is true or false. $F\subset D$

Express the following in set-builder notation:
a) The set A of natural numbers divisible by 3.
b) The set B of pairs (a,b) of real numbers such that ${\displaystyle a+b}$ is an integer.
c) The open interval ${\displaystyle C=(-2,2)}$.
d) The set D of 20 element subsets of N.

Is ${\displaystyle 2\in \left\{2\right\}}$?

Prove or Disprove: For all sets ${\displaystyle A\text{and}B,|A\cup B|=\left|A\right|+\left|B\right|}$

Discrete math.
Prove the following fact by induction:
For every non-negative integer n, 2 evenly divides ${\displaystyle n^2-7n+4}$.

Prove by mathematical induction: ${\displaystyle \mathrm{\forall }n\ge 1,1^3+2^3+3^3+\cdots +n^3=\frac{n^2{(n+1)}^2}{4}}$

Give the first six terms of the following sequences. You can assume that the sequences start with an index of 1. Logs are to base 2. Indicate whether the sequence is increasing, decreasing, non-increasing, or non-decreasing. The sequence may have more than one of those properties.
a) The first two terms in the sequence are 1. The rest of the terms are the sum of the two preceding terms plus 1.
b) The n-th term is ${\displaystyle n^3}$.
c) The n-th term is ${\displaystyle 2n-5}$.

Please assist with this probability question for discrete mathematics.
A class with n kids lines up for recess. The order in which the kids line up is random with each ordering being likely. There are two kids in the class named Betty and Mary. The use of the word "or" in the description of the events, should be interpreted as the inclusive or. That is "A or B" means that A is true, B is true, or both A and B are true.
What is the probability that Betty is first in line or Mary is last in line as a function of n? Simplify your final expression as much as possible and include an explanation of how you calculated this probability.

Discrete Math
Use a truth table to determine if the given argument form is valid.
${\displaystyle p\to q}$
${\displaystyle r\to p}$
${\displaystyle q\vee r}$
${\displaystyle \therefore q}$

Write the answer using summation or product notation
${\displaystyle (1^3-1)-(2^3-1)+(3^3-1)-(4^3-1)+(5^3-1)}$

Prove using mathematical induction:
${\displaystyle \sum _{k=1}^n{2}^{k-1}=2^n-1}$

How many elements will be there for ${\displaystyle A\times B}$ if ${\displaystyle A=\{a,b,c,d\}}$ and ${\displaystyle B=\{1,2,3,4\}}$
Select one:
1) 6
2) 2
3) 8
4) 16

A person borrowed ${\displaystyle \mathrm{\$}4000}$ on a bank credit card at a nominal rate of ${\displaystyle 24\mathrm{\%}}$ per year, which is actually charged a rate of ${\displaystyle 2\mathrm{\%}}$ per month.
a) what is the effective annual percentage rate (Effective APR) for the card?
b) Assume that the person does not place any additional charges on the card and pays the bank ${\displaystyle \mathrm{\$}300}$ each month to pay off the loan. Let ${\displaystyle B\left(n\right)}$ be the balance owed on the card after n months. Find explicit formula for ${\displaystyle B\left(n\right)}$.
c) How long will be reguired to pay off the debt?

Discrete math
Let ${\displaystyle A=\{2,4,5,7,9\}}$. Choose the correct statement from the below.
Select one:
1) ${\displaystyle 2\subset A}$
2) ${\displaystyle 5\in A}$
3) ${\displaystyle 1\in A}$
4) A is not a set.

Consider the following statement:
” A is a subset of B. Therefore A is a subset of P(B).”
This statement is incorrect as written. Assuming the first sentence is true, what is incorrect about the second sentence? State the second sentence correctly.

HELP needed on HW Questions ASAP
Discrete Mathematics:
11. E is the relation defined on Z as follows: for all ${\displaystyle m,n,\in Z,mEn\iff 4\mid (m-n)}$.
a) Prove that E is equivalence relation.
b) List five elements in [5].
c) Find a partition of set Z based on relation E.

This is for Discrete Math.
Find the number of four non-negative integer ordered tuples
${\displaystyle (x_1,x_2,x_3,x_4)}$
such that
${\displaystyle x_1+x_2+x_3+x_4=16}$
and
${\displaystyle x\le 6,x_2\le 6,x_3\le 6}$ and ${\displaystyle x_4\le 6}$
1) 149
2) 218
3) 312
4) 406
5) 242

Find a formula for the sequence given by
${\displaystyle x_{n+1}-4x_n-5x_{n-1}+2x_{n-2}n\ge 2,}$
with ${\displaystyle x_0=2,x_1=7}$ and ${\displaystyle x_2=15}$

Prove that the compound propositions ${\displaystyle p\wedge (q\to r)}$ and ${\displaystyle urc\text{or}\ne r(p\to (q\wedge urc\text{or}\ne rr))}$ are equivalent by using the Equivalence and Implication Rule and constructing two deductive sequence proofs.

Prove that discrete math: $\mathrm{\forall }x\in R$
$|x-1|=\{\begin{array}{ll}x-1,& ifx\ge 1\\ 1-x,& ifx<1\end{array}$