(P v Q) ^ (P⟶R) ^ (Q⟶S) is equivalent to… i dont quite get this also i need to show a truth table. just explanation will do so i can understand.

Question

(P v Q) ^ (P⟶R) ^ (Q⟶S) is equivalent to…

1. S v R
2. S ^ R
3. S ⟶ R
4. All of above

Answer 1

(P or Q) and (P implies R) and (Q implies S)

First we substitute (P implies Q) with (not P or Q). If you are not sure why we can make this substitution, you can prove these expressions are equivalent/identical using a truth table. Do the same for (Q implies S) and we get the following:

(P or Q) and (not P or R) and (not Q or S)

The "or" and "and" operators are associative and communitive, so we can add parenthesis here:

(P or Q) and ((not P or R) and (not Q or S))

Now we distribute. i.e. A and (B or C) == (A and B) or (A and C); A or (B and C) == (A or B) and (A or C). If we do this carefully we get:

(P and ((not P or R) and (not Q or S))) or (Q and ((not P or R) and (not Q or S)))

(P and (not P or R) and (not Q or S)) or (Q and (not Q or S) and (not P or R))

Distribute again:

(((P and not P) or (P and R)) and (not Q or S)) or (((Q and not Q) or (Q and S)) and (not P or R))

Any time we get (P and not P) we can reduce this to "False" since it's a contradiction. Similarly, anytime we get (P or not P) we can replace it with "True" since it's a tautology. So:

((False or (P and R)) and (not Q or S)) or ((False or (Q and S)) and (not P or R))

Also recall that (False or X) == X; (False and X) == False; (True or X) == True; (True and X) == X. So:

((P and R) and (not Q or S)) or ((Q and S) and (not P or R))

I think this is as simplified as we can get:

(P and R and (Q implies S)) or (Q and S and (P implies R))

I think this is as simplified as we can get.

Drawing a truth table from this point shows 5 possible truth:

1. (P and R and Q and S) or
2. (P and R and not Q and S) or
3. (P and R and not Q and not S) or
4. (not P and R and Q and S) or
5. not P and not R and Q and S)

It doesn't seem to simplify into something nice like the potential answers you mentioned.

Maybe the teacher meant (P and Q) and ... which would simplify to all four variables needing to be true. OR maybe they meant (P or Q) or (...) or (...) which would simplify into R or S rather easily.