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

JM

The statement you are trying to prove is

PC

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.