Discrete Mathematics Answers

Given that D={a, b, c, d} and R is the relation of D that has the matrix D= 0 0 0 1

1 0 0 0

1 0 1 1

1 1 0 1

Find the relation R and the digraph of R

1. If the truth value of P ⊕ Q in 2^3 is 0 0 1 1 1 1 0 0 then what is P <=> Q ins 2^3?
2. If the truth value for P v Q (P or Q) in 2^3 is: 1 1 1 1 1 1 0 0 then what is the truth value for P => Q in 2^3?
3. List all the Premises and Conclusion in the following argument: Either Alfred or Bill (or both) will go to the party. If Bill goes and Claire does not then Dinah will go. Claire will go if Alfred does not go. Therefore Dinah will go.
4. List all the Propositions in the following argument: Either Alfred or Bill (or both) will go to the party. If Bill goes and Claire does not then Dinah will go. Claire will go if Alfred does not go. Therefore Dinah will go.
5. Translate the logical expression into English. P: Today is Thursday. Q: Tomorrow is Saturday. R: I will get paid. ~(P ∧ Q) => ~ R

State the value of x after the statement if P(x) then x:=1 is executed, where P(x) is the statement

“x > 1,”

Given that thevalue of x when thisstatement is reachedis [3 marks]

a) x=0

b) x=1

c) x =2

A) Let A={1,2,3,4}A={1,2,3,4} and R a relations on A whose matrices is M_R = ⎡⎣⎢⎢⎢1010111000101011⎤⎦⎥⎥⎥[1101010011110001],                      
1) Show that (A,R)(A,R) is a poset                        (5 pts) 2) Find maximal, minimal, least and greatest if they exist    (4 pts)

  Determine whether each of the following statements is true or false. State the reason.

i) ∅ ∉ ∅

ii) ∅ ⊆ ∅

iii) {∅} ⊆ ∅

a_n=3a_n-1-4,a₀=1

Use a short truth table to determine whether or not the following argument is valid.

(K & ~C) -> ~(P & R)

J -> (K & P)

A -> (P & R)

Thus, (A & J) -> C

Question 1 options:

Valid

Invalid, with R = false, P = true

Invalid, with A = true, K = true

Invalid, with J = true, R = true

Invalid, with P = true, C is false

Use a short truth table to determine whether or not the following argument is valid.

E -> J

B -> Q

D -> (J & ~Q)

Thus, (E & B) -> D

Question 2 options:

Valid

Invalid, with J is false, Q is true.

Invalid, with D is true, B is false.

Invalid, with D is false, B is true.

Invalid, with E is false, Q is true.

Use a short truth table to determine whether or not the following argument is valid.

~S -> ~(Q v G)

(Q v S) & (G v ~N)

(N v ~S) -> L

So, S & ~N

Question 3 options:

Valid

Invalid when L is true, S is false.

Invalid when N is false, S is false.

Invalid when Q is false, S is true.

Invalid when S is true, G is false.