Discrete Mathematics Answers

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|

Let Q(x, y) be the statement “x + y = x − y.” If the domain for both variables consists of all integers, what are the truth values? 

. Prove that if n is a perfect square, then n + 2 is not a

perfect square.

For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other

Solve each of the following problems. Follow Polya's four stages of problem solving in your solution.

3. A palindromic number or numeral palindrome is a "symmetrical" number like 16461, which remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters. How many palindromes are there between 0 and 1000?

4. Matt is half as old as James will be when Matt is twice as old as James is now. In five years, the sum of Matt's and James' ages will be 100. How old are Matt and James now?

5. Find the digit 120 places to the right of the decimal point in the decimal representation of 9/13

With the help of Venn diagram and membership table prove:

(AUB)-C = (A-C)U(B-C)