Discrete Mathematics Answers

Find a counterexample, if possible, to these universally quantified statements, where the domain

for all variables consists of all integers.

a) ∀x (x2 ≥x)

b)∀x(x>0∨x<0) c) ∀x (x = 1)

What are the truth values of these statements?

[3 marks]

a) ∃!xP(x)→∃xP(x)

b)

∀x P(x) → ∃!xP(x)

c)

∃!x¬P(x)→¬∀xP(x)

    In a group of 35 ex-athletes, 17 play golf, 20 go cycling, and 12 do yoga. Exactly 8 play golf and go cycling, 8 play golf and do yoga, 7 go cycling and do yoga, and 4 do all three activities. How many of the ex-athletes do none of these activities? 

Show your solution.

1.     Show, by the use of the truth table/matrix, that the statement (p∨q)∨ (¬q) is tautology.

2.     Show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent.

Let P(x): x² =x

                  2

Find the following then identify their truth values.

1.  P (1)

2.  P (2)

3.   ∀n,P(n)

4.  Ǝn,P(n)

Identify if the following statements are predicate logic. Give a domain of discourse for each propositional function. (3 items x 5 points)

1. The movie won the Academy Award for the past 2 years.
2. 1 + 3 = 4
3. (x+2)² is a prime number.

Find the inverse of the given function. Upload solution in next number.

Let f:R→R , f(x)= 4x + 3

a. f^-1 (y)= 

b. f^-1 (35) = 

c. f^-1 (-9) = 

Let A be the set of words containing the letter s, and let B be

the set of words containing the letter t.

Express the following set as a combination of sets A and B.

a. The set of words that do not contain the letter s

b. The set of words containing the letter s and the letter t

c. The set of words containing an s, but not a t

d. The set of words that do not contain the letters s and t

e. The set of words containing the letter s or t, but not both

In a group of 35 ex-athletes, 17 play golf, 20 go cycling, and 12 do yoga. Exactly 8

play golf and go cycling, 8 play golf and do yoga, 7 go cycling and do yoga, and 4 do

all three activities. How many of the ex-athletes do none of these activities?

show that p⟷q and (p^q) V (¬p^¬q) are equivalent.