Explore Discrete Math Examples

Discrete math truth tables.
Calculate according to truth tables:
$(0\Rightarrow 1)$ $\Rightarrow$ $[0\wedge \mathrm{\neg <!--\; \neg \; -->}(1\vee 0)]$
How should I solve this?

Negate the statement and express your answer in a smooth english sentence. Hint first rewrite the statement so that it does not contain an implication. The statement is: If the bus is not coming, then I cannot get to school.

Permutations: Discrete Math
How many permutations are there of the set (a,b,c,d,e,f,g)

Can someone help me prove that:
$((A\cap <!--\; \cap \; -->B)\oplus <!--\; \oplus \; -->A{)}^{\prime }=A^{\prime }\cup <!--\; \cup \; -->B$
A′ is A complement.

Discrete math relation properties
I am working on a homework assignment and I am having trouble understanding the problem. I feel as if my professor forgot part of the problem, but I would just like to double check and make sure I am not reading the problem incorrectly. This is the problem:
For $\le <!--\; \le \; -->$ relation on the set of integers, specify if $\le <!--\; \le \; -->$ is Reflexive(R), Antireflexive(AR), Symmetric(S), Antisymmetric(AS), or Transitive(T). Show your analysis.

Discrete math prove sets
I am wondering if my answer to this problem is correct.
Statement: $\mathrm{\forall <!--\; \forall \; -->}A,B,C:[A\subseteq <!--\; \subseteq \; -->B\subseteq <!--\; \subseteq \; -->C]\wedge <!--\; \wedge \; -->[C\times <!--\; \times \; -->B\subseteq <!--\; \subseteq \; -->B\times <!--\; \times \; -->A]⟹<!--\; ⟹\; -->[A=B=C]$.
Question: is the above statement true or false?
Answer: true.
Proof: $x\in <!--\; \in \; -->B\times <!--\; \times \; -->Aandx\in <!--\; \in \; -->C\times <!--\; \times \; -->B$ for some $a\in <!--\; \in \; -->Aandb\in <!--\; \in \; -->B$ which gives us $a\in <!--\; \in \; -->A\subseteq <!--\; \subseteq \; -->B\subseteq <!--\; \subseteq \; -->C\Rightarrow <!--\; \Rightarrow \; -->a\in <!--\; \in \; -->B$ and $a\in <!--\; \in \; -->C$ also we get $b\in <!--\; \in \; -->B\subseteq <!--\; \subseteq \; -->C\Rightarrow <!--\; \Rightarrow \; -->b\in <!--\; \in \; -->C$ så we get $x=(a,b)\in <!--\; \in \; -->C\times <!--\; \times \; -->B$ and $x\in <!--\; \in \; -->B\times <!--\; \times \; -->A$.
Is this proof enough or even correct?
Can someone tell me if I'm right or wrong?

If $b\equiv <!--\; \equiv \; -->3(\mathrm{mod}12)$, what is $8b(\mathrm{mod}12)$?
We can write $b=12q+3$. Then with some algebra $8b=12(8q+2)$.
This is causing me problems because there is not number plus $12(8q+2)$ to tell us the remainder. Is it just zero?

I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test?
1. (b) Show that the 'rule' $g:Z_6\to <!--\; \to \; -->Z_9$ defined by $f([a{]}_6)=[4a{]}_9$ is not a well-defined function.
2. Define a function $f:N\times <!--\; \times \; -->N\to <!--\; \to \; -->N$ by $f((a,b))=gcd(a,b)$
(a) show that f is not one-to-one
(b) show that f is onto
3. Let A, B, C be non-empty sets and let $f:A\to <!--\; \to \; -->B$ and $g:B\to <!--\; \to \; -->C$ be functions.
(a) Show that it $g\circ <!--\; \circ \; -->f$ is onto, then g is onto
(b) Find an example of functions f and g such that $g\circ <!--\; \circ \; -->f$ is onto but where f is not onto

Discrete Math On Recurrence
1. Suppose that a geometric sequence starts with and satisfies the recurrence $a_n=r{a}_{n-<!--\; -\; -->1}$ for every positive integer n.
a) Show that $a_n=a_{0}rⁿ$
b) Find the 100th number in the sequence 3,6,12,24,48, … .
I know this is a another recurrence problem but not sure now to start with this one.

Discrete Math Logic Homework
Consider the following statement:
$\mathrm{\forall <!--\; \forall \; -->}\mathrm{ϵ<!--\; ϵ\; -->}>0,\mathrm{\exists <!--\; \exists \; -->}\mathrm{\delta <!--\; \delta \; -->}>0:(|x-<!--\; -\; -->a|<<!--\; <\; -->\mathrm{\delta <!--\; \delta \; -->}⟹<!--\; ⟹\; -->|f(x)-<!--\; -\; -->L|<<!--\; <\; -->\mathrm{ϵ<!--\; ϵ\; -->}).$
(a) Write the converse of the statement.
(b) Write the contrapositive of the statement.

Show that if $S1\subseteq <!--\; \subseteq \; -->S2$, then $\stackrel{¯<!--\; ¯\; -->}{S_2}\subseteq <!--\; \subseteq \; -->\stackrel{¯<!--\; ¯\; -->}{S_1}$ (the complement of S2 is the subset of the complement of S1)

Discrete math predicate problem
In this problem, we will be using binary predicates F(x, y), G(x, y), etc. to represent functions f, $g:U\to <!--\; \to \; -->U$, etc., where U is the universe. Thus, F(x, y) holds iff $y=f(x)$, G(x, y) holds iff $y=g(x)$, etc.
1. Write predicate statements that expresses the following facts:
- F represents a function.
- F represents a one-to-one function.
- F represents an onto function.
- F and G represent inverse functions of one another.
- H represents the composition function $f\circ <!--\; \circ \; -->g$.
2. Use binary predicates representing functions to give formal proofs (in the style of Sec 1.6 of the following statements:
- “If f and g are one-to-one functions, then so is $f\circ <!--\; \circ \; -->g$.”
- “If f and g are onto functions, then so is $f\circ <!--\; \circ \; -->g$.”

Simple math question:
How many ways are there to choose 12 cookies if there are 5 varieties?
It was wrong in my homework after I tried $5^{12}$.

Discrete math sets help?
How would I do this question?
Suppose $U=\{1,\mathrm{2...},9\}$, $A=$ all multiples of 2, $B=$ all multiples of 3, and $C=\{3,4,5,6,7\}$. Find $C-<!--\; -\; -->(B-<!--\; -\; -->A)$.

Discrete Math Functions
$f:\mathbb{N}\to <!--\; \to \; -->\mathbb{N}$ such that $f(x)=2x$
$f:\mathbb{Z}\to <!--\; \to \; -->\mathbb{Z}$ such that $f(x)=2x$
How are these two different?
And also $h:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ where $h(x)=\sqrt{x}$
$f:\mathbb{N}\to <!--\; \to \; -->\mathbb{N}$ where $f(x)=\sqrt{n}$

If $a<x$ and $b<y$, then the rectangle with corners at (x,y), (x,0), (0,y), and (0,0) has an area greater than ab.

Properly Translate Propositional Statement to Discrete Math
What is the proper way to write this statement mathematically? "All people are smokers or non-smokers but not both."
$H=humans,S=smoker,NS=non-<!--\; -\; -->smoker$

Discrete Math, anagram combinatorics
Find the number of anagrams for the word "ALIVE" so that the letter "A" is before the letter "E" or the letter "E" is before the letter "I". By before we mean any letter previous, not just immediately before.
Any help, totally stumped on this question.

There are 9 distinct chairs. How many ways are there to group these chairs into 3 groups of 3?

Simple discrete math proof.
Prove that if $a,b\in <!--\; \in \; -->{\mathbb{Z}}^{+}$, that a|b and b|a, then $a=b$