Discrete Mathematics Answers

You have given a function λ:R-> R with the following properties (x ∈ R, n∈ N) :
Λ(n) =0, λ(x+1)= λ(x), λ(n+1/2)=1
Find two functions p,q:R-> Rwith q(x) not equal to 0 for all x such that λ(x)= q(x)(p(x)+1)

You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N): λ(n) = 0 , λ(x + 1) = λ(x) , λ (n +1/2)=1
Find two functions p, q : R → R with q(x) not equal to 0 for all x such that λ(x) = q(x)(p(x) + 1).​

There are some people (more than 1 person) in a party. Prove that 2 of them have the same number of friends in the party ( Hint: friendship is a mutual relation)

11 integers are arbitrarily chosen. Prove that 2 of them have the same unit digit

Consider the following sequence of successive numbers of the 2k -th power: 1, 2^2k, 3^2k, 4 ^2k, 5 ^2k, ... Show that the difference between the numbers in this sequence is odd for all k ∈ N.

“The product of two odd numbers is odd.”

Using Direct proof method

The sum of two even numbers is even.

Using Direct proof method

make a truth table
(~p∧q) ∨ (p∧~q)

Statement: “If n is any even integer, then ”

Using Direct proof method