Tell whether each proposition is true, is false, or has unknown status at this time.

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September 22nd 2012, 01:46 PM
kmjt

Tell whether each proposition is true, is false, or has unknown status at this time.

p is true
q is false
r's status is unknown at this time

Tell whether each proposition is true, is false, or has unknown status at this time.

q --> r (I know this would be true because if the hypothesis is false, the truth value is always true)

(p^r) <--> r (The textbook says the answer is true but how would I know this? All I know is that p is true, so if I don't know r it can be either true or false)

(q or r) <--> r (The textbook says this is true but I don't get how. All I know is q is false)

Please help :(
September 22nd 2012, 01:53 PM
Plato

Re: Tell whether each proposition is true, is false, or has unknown status at this ti

Quote:

Originally Posted by kmjt

p is true
q is false
r's status is unknown at this time

Tell whether each proposition is true, is false, or has unknown status at this time.
q --> r (I know this would be true because if the hypothesis is false, the truth value is always true)
(p^r) <--> r (The textbook says the answer is true but how would I know this? All I know is that p is true, so if I don't know r it can be either true or false)
(q or r) <--> r (The textbook says this is true but I don't get how. All I know is q is false)

$p \wedge r \Leftrightarrow r$ ia always true because if $r\equiv T$ both sides are true and if $r\equiv F$ both sides are false.
September 22nd 2012, 06:02 PM
kmjt

Re: Tell whether each proposition is true, is false, or has unknown status at this ti

Quote:

Originally Posted by Plato

$p \wedge r \Leftrightarrow r$ ia always true because if $r\equiv T$ both sides are true and if $r\equiv F$ both sides are false.

I do not quite understand :(
September 22nd 2012, 06:23 PM
Soroban

Re: Tell whether each proposition is true, is false, or has unknown status at this ti

Hello, kmjt!

Quote:

$p\text{ is true.}$
$q\text{ is false.}$
$r\text{ is either true or false.}$

Tell whether each proposition is true, is false, or has unknown status at this time.

$q \to r$
I know this would be true because if the hypothesis is false, the truth value is always true.
Right!

Quote:

$(p\wedge r)\;\leftrightarrow\;r$
The textbook says the answer is true but how would I know this?
All I know is that p is true, so if I don't know r, it can be either true or false.

Since $p$ is true, we have: . $(T \wedge r)\;\leftrightarrow\;r$

If $r$ is true, we have: . $(T \wedge T) \leftrightarrow T \quad\Rightarrow\quad T \leftrightarrow T$

If $r$ is false, we have: . $(T \wedge F) \leftrightarrow F \quad\Rightarrow\quad F \leftrightarrow F$

Quote:

$(q \vee r) \leftrightarrow r$
The textbook says this is true but I don't get how. All I know is q is false

Since $q$ is false, we have: . $(F \vee r) \;\leftrightarrow\; r$

If $r$ is true, we have: . $(F \vee T) \leftrightarrow T \quad\Rightarrow\quad T \leftrightarrow T$

If $r$ is false, we have: . $(F \vee F) \leftrightarrow F \quad\Rightarrow\quad F \leftrightarrow F$