Can anyone help me with this symbolic logic problem?

Question

I wasn't able to solve this problem on my test today, but it's driving me crazy... so I was hoping that someone could help me.

I don't have a way to write out some symbols so I'll use "=" for "if & only if" & ">" for "if & only if".

Premise:
(L=Q)>L
Conclude:
LvQ

I've tried both forms of exportation on the premise, along with using implications, etc. Any suggestions?

Error - I meant both forms of equivalence.

Answer 1

The simplest way to do this is, I think, to translate the implications into disjunctions, and solve from there. For instance, we might have:

(L↔Q)→L
¬(L↔Q)∨L (def. of →)
¬((L→Q) ∧ (Q →L)) ∨ L (def. of ↔)
¬((¬L∨Q) ∧ (¬Q∨L)) ∨ L (def. of →)
(¬(¬L∨Q) ∨ ¬(¬Q∨L)) ∨ L (de Morgan's laws)
(L∧¬Q) ∨ (Q∧¬L) ∨ L (de Morgan's laws, associativity of ∨)
L ∨ Q ∨ L (conjunction elimination)
L ∨ Q (disjunction elimination)

Of course, one or more of these steps may need to be lengthened, depending on what inference rules are available to you, but this is the general idea.