# Truth value and Logical equivalents

• September 10th 2008, 05:12 PM
jrh1337
Truth value and Logical equivalents
Determine the truth value

1a) $\forall x:(\exists y:(x=y \rightarrow x > y))$
1b) $\exists x:(\forall y:(x=y \rightarrow x > y))$

Logical equivalents?

2a) $\forall x: ( P(x) \vee Q(x) )$ and $(\forall x: P(x)) \vee (\forall x: Q(x))$
2b) $\forall x: ( P(x) \wedge Q(x) )$ and $(\forall x: P(x)) \wedge (\forall x: Q(x))$
2c) $\exists x: ( P(x) \wedge Q(x) )$ and $(\exists x: P(x)) \wedge (\exists x: Q(x))$
2d) $\forall x: ( P(x) \rightarrow Q(x) )$ and $(\forall x: P(x)) \rightarrow (\forall x: Q(x))$

James
• September 10th 2008, 08:43 PM
triclino
if all students in a classroom are tall or blond this does not imply that all are tall or all are Blond so 2a are not equivalent.
if all students are tall and Blond then this is equivalent to all students are blond and all students are tall so 2b are equivalent
if there exist a student who is tall and there exist a student who is Blondie then not necessarily there exists a student who is tall and blond so 2c are not equivalent
if all students are Blond whenever they are tall this does not mean that every student is blond if he is tall so 2d are not equivalent