1. Help with a proof

Hi,

I need some help with a homework problem involving a proof
i think it is pretty simple but I've never done a proof before so
I'm at a loss as to where to even begin with this...

I have to prove the following:

Any help greatly appreciated,
regards

-dc

2. Originally Posted by dogcow
Hi,

I need some help with a homework problem involving a proof
i think it is pretty simple but I've never done a proof before so
I'm at a loss as to where to even begin with this...

I have to prove the following:

Any help greatly appreciated,
regards

-dc
i'm not sure i fully understand the notation you are using. but by the definition of the union, $\displaystyle x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[ (x \in P(x)) \cup ( x \in Q(x))\right]$

3. Originally Posted by Jhevon
i'm not sure i fully understand the notation you are using. but by the definition of the union, $\displaystyle x \in (P(x) \cup Q(x)) \Longleftrightarrow \left[ (x \in P(x)) \cup ( x \in Q(x))\right]$

i didnt mean the "V" to be taken as a union its an disjunction (or)

4. To prove $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \equiv \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ do side at a time.
If $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$ is true then for some t $\displaystyle \left[ {P(t) \vee Q(t)} \right]$ if $\displaystyle P(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {P(x)} \right]$
or if $\displaystyle Q(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {Q(x)} \right]$.
Thus we have $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$.

Say that $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ is true.
Then for some s we have P(s) or for some u we have Q(u).
In the first instance: $\displaystyle P(s)\quad \Rightarrow \quad P(s) \vee Q(s) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad$.
In the second, $\displaystyle Q(u)\quad \Rightarrow \quad Q(u) \vee P(u) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad$
So in either we get $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$

5. Originally Posted by Plato
To prove $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right] \equiv \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ do side at a time.
If $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$ is true then for some t $\displaystyle \left[ {P(t) \vee Q(t)} \right]$ if $\displaystyle P(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {P(x)} \right]$
or if $\displaystyle Q(t)\quad \Rightarrow \left( {\exists x} \right)\left[ {Q(x)} \right]$.
Thus we have $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$.

Say that $\displaystyle \left( {\exists x} \right)\left[ {P(x)} \right] \vee \left( {\exists x} \right)\left[ {Q(x)} \right]$ is true.
Then for some s we have P(s) or for some u we have Q(u).
In the first instance: $\displaystyle P(s)\quad \Rightarrow \quad P(s) \vee Q(s) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad$.
In the second, $\displaystyle Q(u)\quad \Rightarrow \quad Q(u) \vee P(u) \Rightarrow \quad \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]\quad$
So in either we get $\displaystyle \left( {\exists x} \right)\left[ {P(x) \vee Q(x)} \right]$

is this the right way to prove something or do i have to use actual examples? I mean I dont get how this proves it beyond just restating it in a different way. Or are you saying I have to prove each side?

6. Originally Posted by dogcow
is this the right way to prove something or do i have to use actual examples?

Originally Posted by dogcow
I mean I dont get how this proves it beyond just restating it in a different way. Or are you saying I have to prove each side?
I taught formal logic for many years. This is an outline of what complete proof might look like. I did leave out some justifying citations because no two authors agree on what some operations are called. You must go to your textbook for details.

But I must be honest and tell you that it seems that you are not quite sure of what it means to prove a logical proposition.

7. Originally Posted by Plato