Explore Discrete Math Examples

Evaluate the limit , if it exists.
${\displaystyle \underset{h\Rightarrow 0}{lim}\frac{{(x+h)}^3-x^3}{h}}$

Let S be the set of all strings of 0’s and 1’s, and define ${\displaystyle D:S\Rightarrow Z}$ as follows: For all ${\displaystyle S\in S}$, ${\displaystyle D\left(s\right)=}$ the number of 1’s in s minus the number of 0’s in s.
a. Is D one-to-one? Prove or give a counterexample.
b. Is D onto? Prove or give a counterexample.

Find the volume of the solid bounded by the elliptic cylinder ${\displaystyle 4x^2+z^2=4}$ and the planes y = 0 and y = z + 2

Identify the correct statements for the given functions from the set {a, b, c, d} to itself.
a) ${\displaystyle f\left(a\right)=b,f\left(b\right)=a,f\left(c\right)=c,f\left(d\right)=d}$
b) ${\displaystyle f\left(a\right)=b,f\left(b\right)=a,f\left(c\right)=c,f\left(d\right)=d}$
c) ${\displaystyle f\left(a\right)=b,f\left(b\right)=b,f\left(c\right)=d,f\left(d\right)=c}$
d) ${\displaystyle f\left(a\right)=b,f\left(b\right)=b,f\left(c\right)=d,f\left(d\right)=c}$
e) ${\displaystyle f\left(a\right)=d,f\left(b\right)=b,f\left(c\right)=c,f\left(d\right)=d}$
f) ${\displaystyle f\left(a\right)=d,f\left(b\right)=b,f\left(c\right)=c,f\left(d\right)=d}$

2 points) Page Turner loves discrete mathematics. She has 8 ''graph theory'' books, 6 books about combinatorics, and 4 ''set theory'' books.
How many ways can she place her discrete mathematics books on the same shelf in a row if:
a) there are no restrictions.
b) graph theory books are next to each other but the others could be anywhere on the shelf.
c) books are organized by their topic (same kinds are next to each other).

How many strings of six lowercase letters from the English alphabet contain 6(a) the letter a? 6(b) the letters a and b, where a is somewhere to the left of b in the string, with all the letters distinct

Let f(n) be the largest ' divisor of n. Can it happen that x < y but f(x) > f(y)?Give an example or explain why it is possible.

Suppose that there are $n=2^k$ teams in an elimination tournament, where there are n/2 games in the first round, with the $n/2=2^{k-<!--\; -\; -->1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
Would $f(n)=f(n/2)+1$ for $n=2^k$ be a recurrence relation for the question above? would I also have to say $f(1)=1?$?

Discrete math counting principles
In how many ways can a teacher distribute 12 identical science books among 15 students if
1) no student gets more than one book?
2) if the oldest student gets two books but no other student gets more than one book?
Initially, I tried the binomial theorems with the idea that the student gets a book or no book but that is not giving me the correct solution.

Discrete math library homework
I am working on a homework question and I am not sure if I am going about the correct way of getting to the correct answer. I feel as this is a trick question. Here is the question:
In order to keep track of circulation numbers, the library asks you to note on a form, when you leave the library, which combinations of 15 subject areas and of 8 types of material (books, current journals, databases, bound journals, videotapes, microfilm, microfiche, DVDs) you used. How many possible ways are there to fill in a line on the form?
I was thinking of multiplying $15\cdot <!--\; \cdot \; -->8=120$. But for some reason that did not seem correct. I was also thinking of doing ${15}^8$. But that number seemed too large.

There is a contest with 40 Pokemons. There are 18 Pokemons who like to fight in the sky, and 23 who like to fight on ground. Several of them like to fight in water. The number of those who like to fight in the sky and on ground is 9. There are 7 Pokemons who like to fight in the sky and in water, and 12 who like to fight on ground and in water. There are 4 Pokemons who like to fight in the sky, on ground, and in water. How many Pokemons like to fight in water. How do i express this problem with discrete math?

Discrete math number problem
How would I justify the following statement.
Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.
I think this is true.
because for example
$2<3<4<5$
$2\ast <!--\; \ast \; -->4\ast <!--\; \ast \; -->5\ast <!--\; \ast \; -->3=120$
Which one less than 121 a perfect square.
So how would I justify it I did
Let n be a integer
$n(n+1)(n+2)(n+3)+1=(m{)}^2$
But I am not sure how to proceed.

In my Discrete Math course, I encounter a question such as
$\mathrm{\exists <!--\; \exists \; -->}x\mathrm{\forall <!--\; \forall \; -->}yx<y\text{ where }x,y\in <!--\; \in \; -->\mathbb{Z},$,
and I explain it like this: "Let $x=1$ and $y=-<!--\; -\; -->2$. For an x in the universe not all y values make the statement $x<y$ true so the case is 1 is not equal -2" and I think this is sufficient for explaining but I wonder whether there is a way to indicate proof with symbols or mathematically. Thanks in advance.

Prove that if $b^2-<!--\; -\; -->4c<0$, then there is no such $x\in <!--\; \in \; -->R$ such that $x^2+bx+c=0$

Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a sequence of consecutive days during which he studies exactly 24 hours.

Equivalence Relations (Discrete Math)
I'm having trouble with this math problem on equivalence relations. Let X be any subset of the set of positive integers Z. Define a relation ~ on X as follows:
I have reflexive proven, having trouble with transitivity and symmetric.

Duality discrete math problem
The dual of a compound proposition that contains only the logical operators ∨ , ∧ , and ¬ is the compound proposition obtained by replacing each ∨ by ∧ , each ∧ by ∨ , each T by F , and each F by T . The dual of s is denoted by s∗. Find the dual of these compound propositions.
a) $p\vee <!--\; \vee \; -->\mathrm{\neg <!--\; \neg \; -->}q$
I got $\mathrm{\neg <!--\; \neg \; -->}p\wedge <!--\; \wedge \; -->q$
b) $p\wedge <!--\; \wedge \; -->(q\vee <!--\; \vee \; -->(r\wedge <!--\; \wedge \; -->\mathrm{T}))$
My answer was $\mathrm{\neg <!--\; \neg \; -->}p\vee <!--\; \vee \; -->(\mathrm{\neg <!--\; \neg \; -->}q\wedge <!--\; \wedge \; -->r)$
c) $(p\wedge <!--\; \wedge \; -->\mathrm{\neg <!--\; \neg \; -->}q)\vee <!--\; \vee \; -->(q\wedge <!--\; \wedge \; -->\mathrm{F})$
My answer was $(\mathrm{\neg <!--\; \neg \; -->}p\vee <!--\; \vee \; -->q)\wedge <!--\; \wedge \; -->\mathrm{\neg <!--\; \neg \; -->}q$