Discrete Mathematics Answers

Define a binary relation P from R to R as follows: for all real numbers x and y,

(𝑥, 𝑦) ∈ 𝑃 ⇔ 𝑥 = 𝑦^2

. Is P a function? Explain.

4. Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack.

a) {(0, 0), (1, 1), (2, 2), (3, 3)}

b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}

c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}

d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2),(3, 3)}

e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0),(2, 2), (3, 3)}

5. Which relation on the set {1, 2, 3, 4} is an equivalence relation and contain {(1, 2), (2, 3), (2, 4), (3, 1)}.

6. Find the transitive closures of the relation {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}

on the set {1, 2, 3,,4}.

4. Suppose that a statement of the form ∀xP(x) is false. How can this be proved?

3. Give an example of a predicate P(x,y) such that ∃x∀yP(x,y) and ∀y∃xP (x, y) have different truth values.

5. Let p be the proposition “I will solve every question in this assignment” and q be the proposition “I will be fully prepared for upcoming topics” Express each of these as a combination of p and q. a. I will be fully prepared for upcoming topics only if I will solve every question in this assignment. b. I will be fully prepared for upcoming topics and I will solve every question in this assignment. c. Either I will not be fully prepared for upcoming topics or I will notsolve every question in this assignment. d. For me to be fully prepared for upcoming topicsit is necessary and sufficient that I solve every question in this assignment

2. What does it mean for two propositions to be logically equivalent? 

6. (a) Draw the logic circuit for the following expression: AB + A(B+C)

(b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

Complete the truth tables for these two Boolean expressions: (a) Output = A + B (b) Output = A + AB

Given the following Boolean expression on it, draw a logic gate circuit for this function. AB+ C (A+B)