Question

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.
copy your answers from part (a) here
statement 1: statement 2:
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: Define the propositions

Let \(p\) be "Martina is in geometry class" and \(q\) be "Austin is singing". Then the negation of \(p\) is \(
eg p\) and the negation of \(q\) is \(
eg q\).

Step2: Write Statement 1 in symbolic form

The statement "If Martina is not in geometry class, then Austin is singing" is \(
eg p
ightarrow q\).

Step3: Write Statement 2 in symbolic form

The statement "If Austin is not singing, then Martina is in geometry class" is \(
eg q
ightarrow p\).

Step4: Create the truth - table for Statement 1 (\(

eg p
ightarrow q\))

\(p\)

\(q\)

\(

eg p\) | \(
eg p
ightarrow q\) |

\(T\)	\(T\)	\(F\)	\(T\)
\(T\)	\(F\)	\(F\)	\(T\)
\(F\)	\(T\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)	\(F\)

Step5: Create the truth - table for Statement 2 (\(

eg q
ightarrow p\))

\(p\)

\(q\)

\(

eg q\) | \(
eg q
ightarrow p\) |

\(T\)	\(T\)	\(F\)	\(T\)
\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(F\)	\(T\)
\(F\)	\(F\)	\(T\)	\(F\)

Step6: Check for equivalence

We compare the truth - values of \(
eg p
ightarrow q\) and \(
eg q
ightarrow p\) for all combinations of \(p\) and \(q\). Since the truth - values are the same for all combinations of \(p\) and \(q\), the two statements are equivalent.

Answer:

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

\(p\)

\(q\)

Statement 1: \(

eg p
ightarrow q\) | Statement 2: \(
eg q
ightarrow p\) |

\(T\)	\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(T\)	\(T\)
\(F\)	\(T\)	\(T\)	\(T\)
\(F\)	\(F\)	\(F\)	\(F\)

(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 \(q\).