The problem I am working on is:
Determine the truth value of each of these statements if
the domain consists of all integers.a) $∀n(n+1>n)$
b) $∃n(2n=3n)$
c) $∃n(n=−n)$
d) $∀n(3n≤4n)$
The only part I am having difficulty with is part (d). The answer key declares that this statement is true. But isn't it really a false statement? Wouldn't any negative number render this statement false?
Best Answer
Given that the domain of $n$, as stated, is all* $n\in \mathbb{Z}$, then your reasoning is correct and $d$ is indeed false. Negative integers would serve as your counterexample showing the statement is false. So the answer key must be wrong, or there was a typo in the problem set!
If the domain of $n$ were $\mathbb{N}$, and depending on how one defines the natural numbers $\mathbb{N}$: would is any integer $n \geq 0$ (or an integer $n\geq 1$).
Hence, in either case, negative numbers are excluded from the domain of $n\in \mathbb{N}$.
Hence, $(d)$ would be true, if the domain were in fact $n \geq 0$: given ANY $n\in \mathbb{N},\;3n\leq 4n$, since $3\leq 4$ is clearly true.