im pretty sure (a->c) v (b->c) entails (avb)->c but not sure how to prove it

Actually it's the other way round.

For example, let :
A = " n is a multiple of 4 "
B = " n is a multiple of 3 "
C = " n is an even integer "

A → C is true
B → C is false
hence, (A→C) or (B→C) is true, because at least one of the two statements is true.

On the other hand, (A or B ) doesn't entail C.
For example if n = 3, (A or B ) is true, while C is false.

Let prove that [(A or B ) → C] → [(A→C) or (B→C)] :

Suppose [(A or B ) → C].

Then : A → (A or B ) → C, hence : A → C
In the same way, B → C.

Conclusion : [(A or B ) → C] → [(A→C) and (B→C)] → [(A→C) or (B→C)].

Notice that [(A or B ) → C] is equivalent to [(A→C) and (B→C)].

This post was edited by feanur on Jan 30 2017 01:48am

definitely the other way round

but i dont think the explanation above is logical
a or b leading to c are not equivalent to a lead c AND b lead C
that can be the case but do not have to.

a->c and b->c leands to a or b leading to c
not the other way round
and the or is not exclusive there.

you can say if (a leads to c and b also leads to c), that means that either ( a or b ) lead to c
but you cant say
if (a or lead to c that means (leads to) that [(a AND b) are leading to c] or that [ a leads to C and b leats to c]
maybe b doesnt.

so the conclusion above "Conclusion : [(A or B ) → C] → [(A→C) and (B→C)]"
doesnt sound logical to me
but i have never had anything to do with formal logic - so maybe im wrong.

[(A or B ) → C] is false only when C is false whereas (A or B ) is true, that is to say in the following situations :
(1) A is true, B is true, C is false
(2) A is true, B is false, C is false
(3) A is false, B is true, C is false

[(A→C) and (B→C)] is false when (A→C) is false, or when (B→C) is false (or when both happen).

(A→C) is false when :
(1) A is true, B is true, C is false
(2) A is true, B is false, C is false

(B→C) is false when :
(1) A is true, B is true, C is false
(3) A is false, B is true, C is false

agreed

i have a question regarding that:
a or b lead to c.
Isnt it the whole point that you want to prove c is true (not false?) how come you always say C is false?
meaning that if either a or b leading to C is true
then it is enough for either one to be true, while the other can be false

so if C is true
then either
(1) A is true, B is true, C is true
(2) A is true, B is false, C is true
(3) A is false, B is true, C is true
(4) A is false, B is false, C is false

with [(A→C) and (B→C)] C can only be true of A AND B are true.

thanks a lot for the help!!