Question 185765
Let P = today is Monday and Q = today is Wednesday


Now translate: 


a) "If today is Monday, then tomorrow is not Wednesday" translates to "If P then Q" which symbolically looks like *[Tex \LARGE P \Rightarrow \sim Q]



Now fill out a truth table:


<table border="1">
<tr><td>P</td><td>Q</td><td>*[Tex \LARGE \sim Q]</td><td>*[Tex \LARGE P \Rightarrow \sim Q]</td></tr>

<tr><td>T</td><td>T</td><td>F</td><td><font color="red">F</font></td></tr>
<tr><td>T</td><td>F</td><td>T</td><td><font color="red">T</font></td></tr>
<tr><td>F</td><td>T</td><td>F</td><td><font color="red">T</font></td></tr>
<tr><td>F</td><td>F</td><td>T</td><td><font color="red">T</font></td></tr>
</table>



Note: let me know if you need help constructing a truth table



The truth values in red are the values we're going to compare later.



b) "If it is false that today is Monday and tomorrow is not Wednesday" translates to "It is NOT the case that P and not Q are true".

In symbolic form, the expression looks like: *[Tex \LARGE \sim\left(P \wedge \sim Q\right)]




Now fill out a truth table:


<table border="1">
<tr><td>P</td><td>Q</td><td>*[Tex \LARGE \sim Q]</td><td>*[Tex \LARGE P \wedge \sim Q]</td><td>*[Tex \LARGE \sim\left(P \wedge \sim Q\right)]
</td></tr>

<tr><td>T</td><td>T</td><td>F</td><td>F</td><td><font color="red">T</font></td></tr>
<tr><td>T</td><td>F</td><td>T</td><td>T</td><td><font color="red">F</font></td></tr>
<tr><td>F</td><td>T</td><td>F</td><td>F</td><td><font color="red">T</font></td></tr>
<tr><td>F</td><td>F</td><td>T</td><td>F</td><td><font color="red">T</font></td></tr>
</table>



c) 

"Today is not Monday or tomorrow is Wednesday" translates to "Not P or Q" which looks like *[Tex \LARGE \sim P \vee Q]


Now make a truth table

<table border="1">
<tr><td>P</td><td>Q</td><td>*[Tex \LARGE \sim P]</td><td>*[Tex \LARGE \sim P \vee Q]</td>

<tr><td>T</td><td>T</td><td>F</td><td><font color="red">T</font></td></tr>
<tr><td>T</td><td>F</td><td>F</td><td><font color="red">F</font></td></tr>
<tr><td>F</td><td>T</td><td>T</td><td><font color="red">T</font></td></tr>
<tr><td>F</td><td>F</td><td>T</td><td><font color="red">T</font></td></tr>

</table>

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Now comparing the truth values that are marked in red from parts a) to c), we see that the red truth values in tables b) and c) are identical. So this means that statements b) and c) are equivalent.