JM - Jane wins math prize
PM - Pete wins math prize
JC - Jane wins chem
PC - Pete wins chem
Construct a 2^4 = 16 row truth table with all possible values of JM PM JC PC
The first 3 statements are assumptions (respectively they are):
not(JM and PM)
PM or PC
The statement you are trying to prove is
Proving this by the truth table amounts to checking that for every row where all the assumptions are true, the value of PC is true.
Note: You can simplify your truth table by only looking at values of JM = true since that is one of your assumptions.