Usuario:Galindo6/ejercicio 10

21.

Determine the truth value of each of the following statements if the universe of discourse for all variables is the set of all integers.

a) $\forall n$ ( $n^{2}$ $\geq 0$ )

b) $\exists n$ ( $n^{2}$ =2)

c) $\forall n$ ( $n^{2}$ $\geq n$ )

d) $\forall n$ $\exists m$ ( $n^{2}$ $<m$ )

e) $\exists n$ $\forall m$ ( $n<m^{2}$ )

f) $\forall n$ $\exists m$ ( $n+m=0$ )

g) $\exists n$ $\forall m$ ( $nm=m$ )

h) $\exists n$ $\exists m$ ( $n^{2}+m^{2}=5$ )

i) $\exists n$ $\exists m$ ( $n^{2}+m^{2}=6$ )

j) $\exists n$ $\exists m$ ( $n+m=4\land n-m=1$ )

k) $\exists n$ $\exists m$ ( $n+m=4\land n-m=2$ )

l) $\forall n$ $\forall m$ $\exists p$ ( $p=(m+n)/2$ )

A) $\forall n$ ( $n^{2}$ $\geq 0$ ) $\equiv$ V

B) $\exists n$ ( $n^{2}$ =2) $\equiv$ V (en el universo de los reales)

C) $\forall n$ ( $n^{2}$ $\geq n$ ) $\equiv$ F (para los numeros negativos)

D) $\forall n$ $\exists m$ ( $n^{2}$ $<m$ ) $\equiv$ V

E) $\exists n$ $\forall m$ ( $n<m^{2}$ ) $\equiv$ V (para los negativos)

F) $\forall n$ $\exists m$ ( $n+m=0$ ) $\equiv$ V ( seria su negativo)

G) $\exists n$ $\forall m$ ( $nm=m$ ) $\equiv$ V (el 1)

H) $\exists n$ $\exists m$ ( $n^{2}+m^{2}=5$ ) $\equiv$ V (por ej. $({\sqrt {2}})^{2}+({\sqrt {3}})^{2}$ )

I) $\exists n$ $\exists m$ ( $n^{2}+m^{2}=6$ ) $\equiv$ V (por ej. $({\sqrt {2}})^{2}+({\sqrt {4}})^{2}$ )

J) $\exists n$ $\exists m$ ( $n+m=4\land n-m=1$ ) $\equiv$ V

K) $\exists n$ $\exists m$ ( $n+m=4\land n-m=2$ ) $\equiv$ V ( $ej.n=4,m=2$ )

L) $\forall n$ $\forall m$ $\exists p$ ( $p=(m+n)/2$ ) $\equiv$ V