Question

consider statements p and q.
p: austin is singing.
q: martina is in geometry class.
(a) write each statement below in symbolic form using p and q.
descriptive form symbolic form
statement 1: if martina is not in geometry class, then austin is singing.
statement 2: if austin is not singing, then martina is in geometry class.
(b) complete the truth table below. use t for true and f for false. you may find it helpful to add more columns.
p q statement 1: statement 2: copy your answers from part (a) here.
t t
t f
f t
f f
(c) are statement 1 and statement 2 equivalent? why or why not? choose the best answer.
statement 1 and statement 2 are equivalent. this is because the truth value of statement 1 is the same as the truth value of statement 2 for each true - false combination of p and q.
statement 1 and statement 2 are equivalent. this is because the two statements are made from p and q, and any two statements made from p and q are equivalent.
statement 1 and statement 2 are not equivalent. this is because the two statements are different, and different statements cannot be equivalent.
statement 1 and statement 2 are not equivalent. this is because the truth value of statement 1 is different from the truth value of statement 2 for at least one true - false combination of p and q.

Explanation:

Step1: Write symbolic forms

For "If Martina is not in geometry class, then Austin is singing", since $q$ is "Martina is in geometry class" and $p$ is "Austin is singing", the symbolic form is $
eg q
ightarrow p$. For "If Austin is not singing, then Martina is in geometry class", the symbolic form is $
eg p
ightarrow q$.

Step2: Complete truth - table for $

eg q
ightarrow p$
When $p = T$ and $q = T$, $
eg q=F$, so $
eg q
ightarrow p$ is $F
ightarrow T=T$. When $p = T$ and $q = F$, $
eg q = T$, so $
eg q
ightarrow p$ is $T
ightarrow T=T$. When $p = F$ and $q = T$, $
eg q=F$, so $
eg q
ightarrow p$ is $F
ightarrow F=T$. When $p = F$ and $q = F$, $
eg q = T$, so $
eg q
ightarrow p$ is $T
ightarrow F=F$.

Step3: Complete truth - table for $

eg p
ightarrow q$
When $p = T$ and $q = T$, $
eg p=F$, so $
eg p
ightarrow q$ is $F
ightarrow T=T$. When $p = T$ and $q = F$, $
eg p=F$, so $
eg p
ightarrow q$ is $F
ightarrow F=T$. When $p = F$ and $q = T$, $
eg p = T$, so $
eg p
ightarrow q$ is $T
ightarrow T=T$. When $p = F$ and $q = F$, $
eg p = T$, so $
eg p
ightarrow q$ is $T
ightarrow F=F$.

Step4: Check equivalence

We compare the truth - values of the two statements for all combinations of $p$ and $q$. The truth - value of $
eg q
ightarrow p$ is the same as the truth - value of $
eg p
ightarrow q$ for each true - false combination of $p$ and $q$. So the two statements are equivalent.

Answer:

(a) Statement 1: $
eg q
ightarrow p$; Statement 2: $
eg p
ightarrow q$
(b)

$p$

$q$

Statement 1: $

eg q
ightarrow p$ | Statement 2: $
eg p
ightarrow q$ |

$T$	$T$	$T$	$T$
$T$	$F$	$T$	$T$
$F$	$T$	$T$	$T$
$F$	$F$	$F$	$F$

(c) Statement 1 and Statement 2 are equivalent. This is because the truth value of Statement 1 is the same as the truth value of Statement 2 for each true - false combination of $p$ and $q$.