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

1. ## 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)

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

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

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

Originally Posted by Plato
$\displaystyle p \wedge r \Leftrightarrow r$ ia always true because if $\displaystyle r\equiv T$ both sides are true and if $\displaystyle r\equiv F$ both sides are false.
I do not quite understand

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

Hello, kmjt!

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

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

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

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

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

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

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

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

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

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