# Usuario:Galindo6/ejercicio 10

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• Ejercicicos 1.3, pagina35

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

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

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

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

e) ${\displaystyle \exists n}$${\displaystyle \forall m}$(${\displaystyle n)

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

g) ${\displaystyle \exists n}$${\displaystyle \forall m}$(${\displaystyle nm=m}$)

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

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

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

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

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

## SOLUCION

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

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

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

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

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

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

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

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

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

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

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

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

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