Sure, here are the answers to your questions: Question 1: a) P(orange) = True, because "orange" contains the letter "a". b) P(lemon) = False, because "lemon" does not contain the letter "a". c) P(true) = True, because "true" contains the letter "a". d) P(false) = True, because "false" contains the letter "a". Question 2: a) If x=0, then P(x) is False, so x remains 0. b) If x=1, then P(x) is False, so x remains 1. c) If x=2, then P(x) is True, so x becomes 1. Question 3: a) ∃xP(x) = "There exists a student who spends more than five hours every weekday in class." b) ∀xP(x) = "Every student spends more than five hours every weekday in class." c) ∃x¬P(x) = "There exists a student who does not spend more than five hours every weekday in class." d) ∀x¬P(x) = "No student spends more than five hours every weekday in class." Question 4: a) Q(1) = False, because 1+1 is not greater than 2*1. b) ∃xQ(x) = True, because there exists an integer x (for example, -1) for which x+1 is greater than 2x. c) ∀xQ(x) = False, because not for all integers x, x+1 is greater than 2x. d) ∃x¬Q(x) = True, because there exists an integer x (for example, 1) for which x+1 is not greater than 2x. e) ∀x¬Q(x) = False, because not for all integers x, x+1 is not greater than 2x. Question 5: a) ∃x(x^2 =2) = True, because there exists a real number x (for example, sqrt(2)) for which x^2 equals 2. b) ∃x(x^2 =−1) = False, because there is no real number x for which x^2 equals -1. c) ∀x (x^2 + 2 ≥ 1) = True, because for all real numbers x, x^2 + 2 is greater than or equal to 1. d) ∀x (x^2 =x) = False, because not for all real numbers x, x^2 equals x. Question 6: a) ∀x (x^2 ≥x) = False, counterexample is x=-2. b) ∀x(x0∨x

Question

Sure, here are the answers to your questions:

Question 1:
a) P(orange) = True, because "orange" contains the letter "a".
b) P(lemon) = False, because "lemon" does not contain the letter "a".
c) P(true) = True, because "true" contains the letter "a".
d) P(false) = True, because "false" contains the letter "a".

Question 2:
a) If x=0, then P(x) is False, so x remains 0.
b) If x=1, then P(x) is False, so x remains 1.
c) If x=2, then P(x) is True, so x becomes 1.

Question 3:
a) ∃xP(x) = "There exists a student who spends more than five hours every weekday in class."
b) ∀xP(x) = "Every student spends more than five hours every weekday in class."
c) ∃x¬P(x) = "There exists a student who does not spend more than five hours every weekday in class."
d) ∀x¬P(x) = "No student spends more than five hours every weekday in class."

Question 4:
a) Q(1) = False, because 1+1 is not greater than 2*1.
b) ∃xQ(x) = True, because there exists an integer x (for example, -1) for which x+1 is greater than 2x.
c) ∀xQ(x) = False, because not for all integers x, x+1 is greater than 2x.
d) ∃x¬Q(x) = True, because there exists an integer x (for example, 1) for which x+1 is not greater than 2x.
e) ∀x¬Q(x) = False, because not for all integers x, x+1 is not greater than 2x.

Question 5:
a) ∃x(x^2 =2) = True, because there exists a real number x (for example, sqrt(2)) for which x^2 equals 2.
b) ∃x(x^2 =−1) = False, because there is no real number x for which x^2 equals -1.
c) ∀x (x^2 + 2 ≥ 1) = True, because for all real numbers x, x^2 + 2 is greater than or equal to 1.
d) ∀x (x^2 =x) = False, because not for all real numbers x, x^2 equals x.

Question 6:
a) ∀x (x^2 ≥x) = False, counterexample is x=-2.
b) ∀x(x>0∨x<0) = False, counterexample is x=0.
c) ∀x (x = 1) = False, counterexample is x=2.

Question 7:
∃x(P(x)∨Q(x)) and ∃xP(x)∨∃xQ(x) are logically equivalent because both statements say that there exists an x for which either P(x) is true or Q(x) is true.

Question 8:
a) ∃!xP(x)→∃xP(x) = True, because if there exists exactly one x for which P(x) is true, then there certainly exists an x for which P(x) is true.
b) ∀x P(x) → ∃!xP(x) = False, because even if P(x) is true for all x, it does not mean that there exists exactly one x for which P(x) is true.
c) ∃!x¬P(x)→¬∀xP(x) = True, because if there exists exactly one x for which P(x) is not true, then it is not the case that P(x) is true for all x.

Knowee AI · Accepted Answer