Let Q(x)Q(x) be the statement “x + 1 > 2xx+1>2x ”. If the domain consists of all integers, let us find the following truth values. a) Since 1>0,1>0, we get that 𝑄(0)Q(0) is true. b) Taking into account that it is not true that 0>-2,0>−2, we get that 𝑄(-1)Q(−1) is false. c) Taking into account that it is not true that 2>2,2>2, we get that 𝑄(1)Q(1) is false. d) Since 1>0,1>0, we get that 𝑄(0)Q(0) is true, and hence ∃𝑥𝑄(𝑥)∃xQ(x) is true. e) Taking into account that it is not true that 0>-2,0>−2, we get that 𝑄(-1)Q(−1) is false, and hence ∀𝑥𝑄(𝑥)∀xQ(x) is false. f) Since it is not true that 2>2,2>2, we get that 𝑄(1)Q(1) is false. Therefore, \neg 𝑄(1)¬Q(1) is true, and thus ∃𝑥¬𝑄(𝑥)∃x¬Q(x) is true. g) Since 1>0,1>0, we get that 𝑄(0)Q(0) is true. We conclude that \neg 𝑄(0)¬Q(0) is false, and hence ∀𝑥¬𝑄(𝑥)∀x¬Q(x) is false.
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Let be the statement “ ”. If the domain consists of all integers, let us find the following truth values.
a) Since we get that is true.
b) Taking into account that it is not true that we get that is false.
c) Taking into account that it is not true that we get that is false.
d) Since we get that is true, and hence is true.
e) Taking into account that it is not true that we get that is false, and hence is false.
f) Since it is not true that we get that is false. Therefore, is true, and thus is true.
g) Since we get that is true. We conclude that is false, and hence is false.