Let displaystyle{F}_{{i}} be in the displaystyle{i}^{{{t}{h}}} Fibonacc number, and let n be ary positive eteger displaystylege{3}Prove thatdisplaystyle{F}_{{n}}=frac{1}{{4}}{left({F}_{{{n}-{2}}}+{F}_{{n}}+{F}_{{{n}+{2}}}right)}

Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where ${\displaystyle (a,b)\in R}$ if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.

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

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: ${\displaystyle \cap \cup }$

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

Let Q(x) be a quantifer for the universe ${\mathbb{Z}}^{+}$. I want to check whether $(\mathrm{\forall }y(Q(y)\vee \mathrm{\exists }xQ(x)))⟺\mathrm{\exists }yQ(y)$.

This is a discrete math (combinatorics and discrete probability) problem. Please explain each step in detail and do not copy solutions from Chegg.
Using the Pigeonhole Principle, prove that in every set of 100 integers, there exist two whose difference is a multiple of 37. Identify the function (including its domain and target) outlined in either of our class resources while explaining how the principle is being applied.

Prove that if $A\subseteq B$, and A is uncountable, then B is uncountable"
I think that for an answer we could reason with cardinality as follows Suppose that B is countable, then $\mid B\mid \le \mid \mathbb{N}\mid$.
But if $A\subseteq B$, then $|A|\le |B|\le |\mathbb{N}\mid$ that is untrue if A is uncountable, since any uncountable set should have higher cardinality then natural numbers.