Proof logic using conditional proof

Question

I have a question where I need to show that $(a \wedge b), (a \vee b)$ entails $(a \rightarrow b) \wedge(c \rightarrow a)$

Here's my proof

$(1) \space (a \wedge b) \qquad $ premise

$(2) \space a \qquad$ supposing for conditional proof

$(3) \space b \qquad$ simplification of $(1)$

$(4) \space a \rightarrow b \qquad$ conditional proof from $(2),(3)$

$(5) \space c \qquad$ supposing for conditional proof

$(6) \space a \qquad$ simplification of $(1)$

$(7) \space c \rightarrow a \qquad$ conditional proof from $(5),(6)$

$(8) \space (a \rightarrow b) \wedge(c \rightarrow a) \qquad$ conjunction of $(4),(7)$

Is this correct? I guess where I'm not too sure of are lines $(2)-(4)$ and $(5)-(7)$ since I don't really use my supposition to get my conditional proof. For example, to me it seems something like: suppose I'm hungry and I know I have to go to school; by above it means If I'm hungry then I have to go to school. Which doesn't really make any sense; maybe it does in logic?

Answer 1

Your proof is correct!

And yes, the conditionals in logic can be a little counterintutive at times, in that they are true as soon as the 'then' part is true. Look up 'Paradox of Material Implication' to see how conditionals in logic don't quite match up with the way we use conditionals in real life.