Hello Gilvanildo

Welcome to Math Help Forum!

I have the answer without truth tables - but see below!This means that either Tom or Roy got the highest grade, and therefore Jen did not.

Since Jen did not get the highest grade, Roy must have got the lowest grade.-if Roy did not get the lowest grade, then Jen got the highest 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 and , the first proposition is , which has the following truth table:

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 . 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 . But each person can only get one grade. So, in addition to these, we have: . But .

But , since there can be only one person with the highest grade. And . So is false.

Finally . So is true, and therefore is false (since ) and therefore is true. So the positions are Tom, Jen, Roy.

Grandad