Determine the truth values of the following propositions. Explain.a) ∅ ∈ {1}b) {1} ∈ {1}c) {1} ⊂ {1}d) {1} ∈ {1, {1}}e) {1} ⊆ {1, {1}}f) {{1}} ∈ 𝒫({1, {1}})

Determine whether each of these statements is true or false.a) 0 ∈ ∅b) ∅ ∈ {0}c) {0} ⊂ ∅d) ∅ ⊂ {0}e) {0} ∈ {0}f) {0} ⊂ {0}g) {∅} ⊆ {∅}

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)

U = {1, 2, {1}, {2}, {1, 2}}      A = {1, 2, {1}}       B = {{1}, {1, 2}}     C = {2, {1}, {2}}.Which one of the following statements is valid if x ∉ B U C? (Hint: Determine U – (B U C).)a.x ∈ {1}.b.x ∈ ⊘.c.x ∈ {1, 2}.d.x ∈ B and x ∈ C.

Let A = {{⊘}, 1, {1}}.Which one of the following alternatives regarding A is FALSE?a.{{1}} ⊂ Ab.{⊘} ⊂ Ac.|Ƥ (A)| = 8d.{⊘} ∈ AClear my choice

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)