# Logic Connectives

• Sep 29th 2009, 09:34 PM
Gilvanildo
Logic Connectives
John would like to find out how his three friends (Tom, Jen, and Roy) did in MATH class. He knows that
-if Tom did not get the highest grade, then Roy did.
-if Roy did not get the lowest grade, then Jen got the highest grade.
Is it possible to figure out the ranking of the three friends? If not, what other information do you need? (Hint: use truth tables)

I've set the variables for all the statement;
p: Tom got the highest grade.
q: Roy got the highest grade.
r: Roy got the lowest grade.
s: Jen got the highest grade.

I've tried all possiblities but I cannot come up with anything.
Thank You...
• Sep 29th 2009, 09:57 PM
Hello Gilvanildo

Welcome to Math Help Forum!

I have the answer without truth tables - but see below!
Quote:

Originally Posted by Gilvanildo
John would like to find out how his three friends (Tom, Jen, and Roy) did in MATH class. He knows that
-if Tom did not get the highest grade, then Roy did.

This means that either Tom or Roy got the highest grade, and therefore Jen did not.
Quote:

-if Roy did not get the lowest grade, then Jen got the highest grade.
Since Jen did not get the highest grade, Roy must have got the lowest grade.

So the order highest to lowest must be Tom, Jen, Roy.

In support of the argument above, you could use a truth table and then some further logical arguments (which again can be verified using truth tables), as follows:

Using your definitions of $\displaystyle p$ and $\displaystyle q$, the first proposition is $\displaystyle \neg p \Rightarrow q$, which has the following truth table:

$\displaystyle \begin{array}{c c||c|c| c} p&q&\neg p& \Rightarrow& q\\ \hline 0&0&1&0&0\\ 0&1&1&1&1\\ 1&0&0&1&1\\ 1&1&0&1&1\\ \hline &&(1)&(3)&(2) \end{array}$

The columns are evaluated in order (1) - (3), the output column therefore being (3). Clearly this is the same as the output of the truth table for $\displaystyle p\lor q$. This, then, proves the first part of my first conclusion, that either Tom or Roy got the highest grade.

Similarly we can write your second expression as $\displaystyle \neg r \Rightarrow s$. But each person can only get one grade. So, in addition to these, we have: $\displaystyle q \Rightarrow \neg r$. But $\displaystyle \Big((q \Rightarrow \neg r)\land(\neg r \Rightarrow s)\Big)\Rightarrow(q \Rightarrow s)$.

But $\displaystyle q \Rightarrow \neg s$, since there can be only one person with the highest grade. And $\displaystyle \Big((q \Rightarrow s) \land (q\Rightarrow \neg s)\Big)\Rightarrow \neg q$. So $\displaystyle q$ is false.

Finally $\displaystyle \Big((p\lor q) \land \neg q\Big) \Rightarrow p$. So $\displaystyle p$ is true, and therefore $\displaystyle s$ is false (since $\displaystyle p\Rightarrow \neg s$) and therefore $\displaystyle r$ is true. So the positions are Tom, Jen, Roy.