Question

Read the two statements shown below.

If the weather is not cold, Meg will go swimming.
The weather is cold, or Meg will go swimming.

Create truth tables for the logical form of the two statements (not to be submitted). Use the truth tables to determine whether the two statements are logically equivalent. Justify your answer.

Answer 1

they are equivalent, as the truth tables are the same

to turn \(P\to Q\) into an equivalent or statement
\[P\to Q\equiv \lnot P\lor Q\]

Answer 2

here is the truth table for \(P\to Q\)
\[\begin{array}{|c|c|c}
P
& Q
& P\to{}Q \\
\hline
T & T &T \\
T & F & F \\
F & T & T \\
F & F & T \\
\hline
\end{array}\]

Answer 3

and here it is for \(\lnot P\lor Q\)
\[\begin{array}{|c|c|c|c}
P
& Q
& \lnot{}P
& \lnot{}P\lor{}Q \\
\hline
T & T & F & T \\
T & F & F & F \\
F & T & T & T \\
F & F & T & T \\
\hline
\end{array}\]

Answer 4

you can see that the last columns are identical, so it is equivalent

Answer 5

Thanks you so much!! So if I wrote this in sentence form would this be correct

These statements are equivalent, as their truth tables are the same. The P is true twice, and false twice for both statements, then the Q is T then F then T then F for both statements. The truth table column for P→Q is T, F, T, T as it is the same for ¬P∨Q.

Answer 6

yes although i would skip
" The P is true twice, and false twice for both statements, then the Q is T then F then T then F for both statements."
that is just how the truth table is constructed
maybe put

Answer 7

?