Discrete Mathematics Answers

8. In the previous problem, put a negation in front of the logical expression for “Someone

in your class is perfect”, then move the negation until negation only appears directly

in front of S(x) or P(x), by applying DeMorgan’s Laws.

9. Let these be the hypotheses:

If it’s not cold or if it’s not windy, then I will go walking.

If I go walking, I’ll feel good.

I don’t feel good.

Use rules of inference to show that the above hypotheses imply the conlusion:“It’s

windy.”

10. Use a proof by contraposition to prove that if m and n are integers and mn is even,

then m is even or n is even.

11. Use a proof by contradiction to prove that if x is an irrational number and y is a rational

number, then x + y is an irrational number.

6. Show that (p∧q) → r and (p → r)∧(q → r) are not logically equivalent, without using

truth tables.

7. Express each of these statements into logical expressions using predicates, quantifiers,

and logical connectives. Let the domain consist of all people. Let S(x) be “x is in your

class,” P(x) be “x is perfect.”

(a) Nobody is perfect. (Use only the universal quantifier.)

(b) Nobody is perfect. (Use only the existential quantifier.)

(c) Nobody in your class is perfect.(d) Not everybody is perfect.

(e) Someone in your class is perfect.

(f) Not everyone in your class is not perfect.

3.State the converse, contrapositive and inverse of the conditional statement: When it’s

hot out, it is necessary that I eat ice cream.

4. Steve, Bill and Larry go to a bar. The bartender asks: “Does everyone want beer?”

Steve says: “I don’t know.” Bill says: “I don’t know.” Finally Larry says: “No, not

everyone wants beer.” The bartender proceeds to serve beer to those among these three

who want it. How did he figure out who wanted beer?

5. Show that ¬p → (q → r) and q → (p∨r) are logically equivalent, in two different ways:

(i) use a truth table, (ii) without using truth tables.

1. Let p and q be the propositions

p: It snowed.

q: Eve goes skiing.

Express each of these propositions using p and q and logical connectives.

(a) It snowed but Eve does not go skiing.

(b) Whenever Eve goes skiing, it snowed.

(c) It having snowed is sufficient for Eve to go skiing.

(d) For Eve to go skiing, it is necessary that it snowed.

(e) If it didn’t snow, then Eve does not go skiing.

(f) Whenever Eve doesn’t go skiing, it has not snowed.

2. For the propositions in Problem 1, identify all pairs of propositions that are logically

equivalent. Justify your answer.

Let P(x) be the statement ”x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in English.

Let W(x, y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence.

Conventional way of writing

Classify the following either mathematical expression (ME) or mathematical sentence (MS)

1. 5>8

2. a+b

3. x+y= a+b

4. t/100

5. 3. 141627980643891545746549

Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f : A → B is both one-to-one and onto.]

f : R × R → R, f(n, m) = 2m − n

Construct a truth table for each of these compound statements."( p \\leftrightarrow" q) "\\to(\\lnot p \\leftrightarrow q )"