What is the probability that a card selected at random from a standard deck of 52 cards is a number 5 or 6 or 10?

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

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

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.

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

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.

Draw the Hasse diagram representing the partial ordering {(a, b) | a divides b} on {1, 2, 3, 4, 6, 8, 12}.

Discrete Mathematics Division Algorithm proof
I'm not quite sure how to do this problem if anyone can do a step by step to help me understand it I would appreciate it a lot.
Let a and b be positive integers with $b>a$, and suppose that the division algorithm yields $b=a\cdot q+r$, with $0\le r<a$. (note: its a zero)
Prove that $\mathrm{l}\mathrm{c}\mathrm{m}(a,b)-\mathrm{l}\mathrm{c}\mathrm{m}(a,r)=\frac{a^2\cdot q}{gcd(a,b)}.$