Let 𝑃(𝑥) denote the statement "𝑥 ≤ 4, " What are the truth values for:a. P(0)b. P(4)c. P(6)

What are the truth values of these statements? a) ∃!xP(x)→∃xP(x) b) ∀x P(x) → ∃!xP(x) c) ∃!x¬P(x)→¬∀xP(x)

Determine whether the value 𝑥=4 makes the following equations true or false.True False 𝑥+48−1=06+𝑥=25−𝑥2=−113(7−𝑥)+8=16

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. [4 marks] a)∃x(x2 =2) b)∃x(x2 =−1) c) ∀x (x2 + 2 ≥ 1) d) ∀x (x2 =x)

What must the truth values of 𝑝𝑝 and 𝑞𝑞 be in order to make the sentence 𝑝→¬𝑞𝑝→¬𝑞 false?

Question 1 LetP(x)bethestatement “the word x contains the letter a.” What are the truthvalues of the following? [4 marks] a) P(orange) b) P(lemon) c) P(true) d) P(false) Question 2 State the value of x after the statement if P(x) then x:=1 is executed, where P(x) is the statement “x > 1,” Given that thevalue of x when thisstatement is reachedis [3 marks] a) x=0 b) x=1 c) x =2 Question 3 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. [4 marks] a) ∃xP(x) b) ∀xP(x) c) ∃x¬P(x) d) ∀x¬P(x) Question 4 Let Q(x) be the statement “x+1>2x”. If the domain consists of all integers, what are these truth values? a) Q(1) b) ∃xQ(x) c) ∀xQ(x) d) ∃x¬Q(x) e) ∀x¬Q(x) [5 marks] Question 5 Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. [4 marks] a)∃x(x2 =2) b)∃x(x2 =−1) c) ∀x (x2 + 2 ≥ 1) d) ∀x (x2 =x) Question 6 Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of allintegers. [4 marks] a) ∀x (x2 ≥x) b)∀x(x>0∨x<0) c) ∀x (x = 1) Question 7 Show that ∃x(P(x)∨Q(x)) and ∃xP(x)∨∃xQ(x) are logically equivalent. [3 marks] Question 8 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)