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)

a) Q(1) translates to "1+1>2*1" which simplifies to "2>2". This statement is false.

b) ∃xQ(x) translates to "There exists an x such that x+1>2x". This statement is true for negative integers and zero. For example, if x=-1, the inequality becomes "-1+1 > 2*-1" which simplifies to "0 > -2". This statement is true.

c) ∀xQ(x) translates to "For all x, x+1>2x". This statement is false because it is not true for all integers. For example, if x=2, the inequality becomes "2+1 > 2*2" which simplifies to "3 > 4". This statement is false.

d) ∃x¬Q(x) translates to "There exists an x such that not (x+1>2x)". This statement is true because there are integers for which x+1 is not greater than 2x. For example, if x=2, the inequality becomes "2+1 > 2*2" which simplifies to "3 > 4". This statement is false, so the negation is true.

e) ∀x¬Q(x) translates to "For all x, not (x+1>2x)". This statement is false because there are integers for which x+1 is greater than 2x. For example, if x=-1, the inequality becomes "-1+1 > 2*-1" which simplifies to "0 > -2". This statement is true, so the negation is false.