Question

Which is logically equivalent to “if today is Sunday Matt cannot play hockey”?
A. Today is Sunday and Matt can play hockey
B. If Matt plays hockey then today is not Sunday
C. Today is Sunday and Matt cannot play hockey
D. Today is not Sunday if and only if Matt plays hockey

Answer 1

Hint: This is a problem related to mathematical reasoning in which we have to specify certain events with a truth value and then proceed to the conclusion of further events.
We must assume our statement as true and display it symbolically and then compare with the best suitable choice amongst the provided options. Hence, this obtained expression will be our answer.

Complete step-by-step answer:

In mathematical reasoning, a sentence or a proposition is an assertive sentence which is either true or false but not both. An assertive sentence is a sentence making an assertion. A true statement is also known as a valid statement and a false statement is an invalid statement. A statement which is true and false simultaneously is called a paradox.
The truth or falsity of a statement is called its truth value.
The given statement can be broken in two parts as follows:
Statement (1): Today is Sunday (S).
 Statement (2): Matt cannot play hockey (H).
Logical equivalence of the problem statement can be elaborated as:
$Equivalence=S\wedge H$.
Assuming the above equivalence to be true.
Now, considering the first option and expressing it in logical expression:
For option (a): $Equivalence=S\wedge \sim H$.
So, this equivalence is not matching with our truth statement.
Now, considering the second option:
For option (b):$\begin{align}
  & Equivalence=\sim H\vee \sim S \\
 & \text{ =}\sim \left\{ \sim \text{(}H\wedge S) \right\} \\
 & \text{ =}H\wedge S \\
\end{align}$.
So, it matches with our truth statement.
Hence, option (b) is correct.

Note: This problem can alternatively be solved without using expression by understanding the meaning of all stated options and then matching each one with truth value. This is an important method in solving multiple-choice problems having 2 or more options, where we can eliminate the incorrect option or check the options for their correctness.