Homework Help: Showing logical equivalence

1. Jun 22, 2012

bonfire09

1. The problem statement, all variables and given/known data
(b) Show that (p → q) ∨ (p→ r) is equivalent to p → (q ∨ r).

2. Relevant equations

the ~ means negate

3. The attempt at a solution

Im not sure if i did this correctly
(p → q) ∨ (P → r)
(~p∨q) ∨ (~p∨r) used the conditional law p→q equivalent to ~p∨q
((~p∨q)∨~p)∨((~p∨q)∨r)) distributive law
(~p∨q)∨(~p∨q)∨r
(~p∨q)∨r
~p∨(r∨q) associative law
p →(q∨r)

2. Jun 23, 2012

tiny-tim

hi bonfire09!

however, after …
… don't you notice that they're all ∨ ,

so you can rearrange them (using the …?… law), and then use ~p∨~p = ~p

3. Jun 23, 2012

HallsofIvy

Are you required to do it that way? Setting up an 8 case "truth table" shows that both statements are false in case p= T, q= r= F, and true in all other cases.

4. Jun 23, 2012

bonfire09

That im not sure upon. In Velleman's book he is not so clear about what he wants us to show. But I think what I did suffices.