Let P be the power set of {a,b,c} Define a function from P to the set of integers as follows: f(A) = |A| . (|A| is the cardinality of A.) Is f injective? Prove or disprove. Is f surjective? Prove or disprove.

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

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

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

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.

If G is n regular, then G has n disjoint perfect matchings.
Let G be a bipartite simple graph show that:
If G is n regular, then G has n paarwise disjoint perfect matchings.
It's firstly easy to show that G has a perfect matching by using Hall’s Theorem, $|N(S)|\ge |S|$ with N the potential mathings and S a arbitary subset of one part of the bipartite graph G. But how to show that there is n perfect matchings and besides disjoint. I've seen a answer by multiplying d to $|N(S)|\ge |S|$, but I don't why should we do this. If anyone could help me with some tipps, I would be very grateful.

If ${\displaystyle x_1=-1,x_2=1,X_n=3X_{(n-1)}-\mid 2X_{(n-2)},\mathrm{\forall }n\ge 3.}$ Find the general term ${\displaystyle X_n}$

1) ${\displaystyle 110001}$ binary number is equivalent to which of the following decimal number
a) ${\displaystyle 48}$
b) ${\displaystyle 49}$
c) ${\displaystyle 59}$
d) ${\displaystyle 58}$
2) Which among the following is the recursive definition of Factorial, i.e., n! ?
a) ${\displaystyle f\left(0\right)=0,f\left(n\right)=\left(n\right)f(n-1),}$ where ${\displaystyle n\in Z}$ and ${\displaystyle n\ge 1}$
b) ${\displaystyle f\left(0\right)=1,f\left(n\right)=(n+1)f(n-1),}$ where ${\displaystyle n\in Z}$ and ${\displaystyle n\ge 1}$
c) ${\displaystyle f\left(0\right)=1,f\left(n\right)=\left(n\right)f(n-1),}$ where ${\displaystyle n\in Z}$ and ${\displaystyle n\ge 1}$
b) ${\displaystyle f\left(0\right)=0,f\left(n\right)=(n+1)f(n-1),}$ where ${\displaystyle n\in Z}$ and ${\displaystyle n\ge 1}$