Discrete Mathematics Answers

Suppose that the domain of the predicate P (x) consists of 1, 2, 3, 4, and 5. Write

out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

Build a Fully truth table and identify if tautology, Contradiction or Contingency

¬(¬(P∧Q)↔(¬P)∨(¬Q))

Make a Fully truth table and Show that they are logically equivalent

⌐ (⌐p ∧ q) ∧ (p ∨q) ≡ p

Prove that n•P(n−1,n−1)= P(n,n).

Prove that for all integer n3, P(n+1,3)−P(n,3)=3P(n,2).

Establish the validity of the argument with the premises p -> (q -> r) , p \/ s , t ->q , ~s and ~r -> ~t

How many subsets of the set {1, 2, 3, 4, 5, 6, 7, 8} do not contain two consecutive integers ?

In a class of n

n students (we consider all students as distinct), we want to make k

k groups where each group must contain at least 1 student. Let S(n,k)

S(n,k) denote the number of ways in which such groups can be formed. Which of the following is/are true? (More than one options may be correct.)

S(n+1,k)=kS(n,k)+S(n,k−1)

S(n+1,k)=kS(n,k)+S(n,k−1)

S(n+1,k+1)=kS(n,k+1)+S(n,k)

S(n+1,k+1)=kS(n,k+1)+S(n,k)

For  n,k>1,S(n,k)=∑

j=1

n−1

(n−1

j

)S(j,k−1)

n,k>1,S(n,k)=∑j=1n−1(n−1j)S(j,k−1)

For  n,k>1,S(n,k)=∑

j=1

n−1

(n

j

)S(j,k−1)

For integers 𝑥 and 𝑦, consider the proposition ∃𝑥∀𝑦 (𝑥 + 𝑦 = 10). Is the proposition true, false, or does it depend on the values of 𝑥 and 𝑦?