Discrete Mathematics Answers

For the proposition(p ∨¬r)∧(¬p∨(q∨¬r)
a. Draw the truth table
b. Build the logic circuit which outputs the given compound proposition from input bits p, q,
and r.

A professor in a discrete mathematics class passes out a form asking students to check all the mathematics and computer science courses they have recently taken. The finding is that out of a total of 50 students in the class, 30 took precalculus; 16 took both precalculus and Java; 18 took calculus; 8 took both calculus and Java; 26 took Java; 47 took at least one of the three courses; and 9 took both precalculus and calculus.
a. How many students did not take any of the three courses?
b. How many students took all three courses?
c. How many students took precalculus and calculus but not Java? How many students took precalculus but neither calculus nor Java?

For each of the following sets, determine its power set.
a. A = {a, b, c, d}
b. A = {{1}, {1, 2}}

Which of the following are partitions of , the set of real numbers? Explain your answers.
a. {In : n ∈ ℤ}, where In = {x ∈ ℝ : n ≤ x ≤ n + 1}
b. {Jn : n ∈ ℤ }, where Jn = {x ∈ ℝ: n ≤ x < n + 1}

Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find
a. A × B × C.
b. C × (B × A).

Let A = {n ∈ Z | n = 5r for some integer r} and B = {m ∈ Z | m = 20s for some integer s}.
a. Is A ⊆ B? Explain.
b. Is B ⊆ A? Explain

Let X = {0, 1, 2}. List the elements of each of the following sets:
a. {z : z = 2x and x ∈ X}
b. {z : z = x + y where x ∈ X and y ∈ X}

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)