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 :(

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)

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

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

Quote:

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 :(

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

Hello, kmjt!

Quote:

$\displaystyle p\text{ is true.}$

$\displaystyle q\text{ is false.}$

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

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!

Quote:

$\displaystyle (p\wedge r)\;\leftrightarrow\;r$

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: .$\displaystyle (T \wedge F) \leftrightarrow F \quad\Rightarrow\quad F \leftrightarrow F$

Quote:

$\displaystyle (q \vee r) \leftrightarrow r$

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: .$\displaystyle (F \vee F) \leftrightarrow F \quad\Rightarrow\quad F \leftrightarrow F$