Question
Determine the truth value of each of these statements if the domain consists of all integers.$$\begin{array}{ll}{\text { a) } \forall n(n+1>n)} & {\text { b) } \exists n(2 n=3 n)} \\ {\text { c) } \exists n(n=-n)} & {\text { d) } \forall n(3 n \leq 4 n)}\end{array}$$
Step 1
This statement is saying that for all integers $n$, $n+1$ is greater than $n$. This is true because if you add 1 to any integer, it is greater than what the integer was originally. So, the statement a) is **True**. Show more…
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Determine the truth value of each of these statements if the domain for all variables consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n\left(n^{2} \geq 0\right)} & {\text { b) } \exists n\left(n^{2}=2\right)} \\ {\text { c) } \forall n\left(n^{2} \geq n\right)} & {\text { d) } \exists n\left(n^{2}<0\right)}\end{array} $$
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Determine the truth value of each of these statements if the domain for all variables consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n \exists m\left(n^{2}<m\right)} & {\text { b) } \exists n \forall m\left(n<m^{2}\right)} \\ {\text { c) } \forall n \exists m(n+m=0)} & {\text { d) } \exists n \forall m(n m=m)} \\ {\text { e) } \exists n \exists m\left(n^{2}+m^{2}=5\right)} & {\text { f) } \exists n \exists m\left(n^{2}+m^{2}=6\right)}\end{array} $$ $$ \begin{array}{l}{\text { g) } \exists n \exists m(n+m=4 \wedge n-m=1)} \\ {\text { h) } \exists n \exists m(n+m=4 \wedge n-m=2)} \\ {\text { i) } \forall n \forall m \exists p(p=(m+n) / 2)}\end{array} $$
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Determine the truth value of each of these statements if the domain consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \quad \exists x\left(x^{3}=-1\right)} & {\text { b) } \exists x\left(x^{4}<x^{2}\right)} \\ {\text { c) } \quad \forall x\left((-x)^{2}=x^{2}\right)} & {\text { d) } \forall x(2 x>x)}\end{array} $$
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