Discrete Mathematics Answers

A. Let p and q be propositions

p: 4 is a rational number

q: 3 is an irrational number.

Express each of these propositions as an English sentence:

1. ¬p =

2. p v q =

3. p ^ q =

4. p→q =

5. p ↔q =

• Let P(x, y) be the statement “student x has taken class y”, where the domain for x consists of all students and y consists of all computer engineering courses at your school.

• Express each of the quantification in English sentences

1. ∀x ∀y P(x,y)

2. ∀x Ǝy P(x,y)

3. Ǝx ∀y P(x,y)

4. Ǝx Ǝy P(x,y)

7. Which of the following statement is not a member of set X?

X = {tiger, lion, puma, cheetah, leopard, cougar, ocelot}

A. cougar

B. bobcat

C. puma

D. Tiger

8. What type of set is H?

A. Empty

B. Finite

C. Infinite

D. None of the above.

9. Which of the ff. sets are finite?

A. {vowels}

B. {days of the week}

C. {primary colors}

D. All of the above.

10. Which of the ff. sets is equal to set P?

P = {Monday, Tuesday, Wednesday, Thursday, Friday}

A. W = {Thursday, Friday, Saturday, Sunday, Monday}

B. X = {Tuesday, Wednesday, Thursday, Friday, Saturday}

C. Y = {Thursday, Friday, Monday, Tuesday, Wednesday}

D. All of the Above.

II - Truth Table (Make a truth table for the given expression.) 10pts each.

11. (~p∧q) ∨ (p∧~q)

12. (~p∧q) ∨ (~p∨q)

III - Set Analysis.

Consider the counting numbers {1,2,3,4,5,6,7,8,9}.

Let A={1,3,5,7,9}, B={2,4,6,8}, C={3,5}

13. Find A union B. {A∪B}

14. Find A complement. {A’}

15. Find B intersect C. B∩C}

Select the statement that is the negation of “All summer days are muggy.”

A. All muggy days are summer.

B. Some summer days are muggy.

C. Some summer days are not muggy.

D. No summer days are muggy.

A. Let p and q be propositions

p: 4 is a rational number

q: 3 is an irrational number.

Express each of these propositions as an English sentence:

1. ¬p =

If X has n elements , how many elements does the power set of X have?

What is the cardinality of each of these sets? a) {a} b) {{a}} c) {a, {a}} d) {a, {a}, {a, {a}}}

eLMS Practice Exercises

• Let P(x, y) be the statement “student x has taken class y”, where the domain for x consists of all students and y consists of all computer engineering courses at your school.

• Express each of the quantification in English sentences

1. ∀x ∀y P(x,y)

2. ∀x Ǝy P(x,y)

3. Ǝx ∀y P(x,y)

4. Ǝx Ǝy P(x,y)

Determine whether each of these functions from Z to Z is one-to-one.

a) f(x) = x−2

b) f(x) = x2 −1

c) f(x) = x3

Find out whether the following functions are one to one or not

(a) f(x)=3x+4 (b) f(x)=x²+4 (c) h(x)=13x⁵+5 (d) g(x)=x⁴+3 (e) p(x)=1/x+3 (f) f(x) =|2x+5|