Predicate Logic Conclusion

Question

I want to show that $ A \vee B $ is a consequence of $ A \wedge B $. So far I have tried some replacement rules so I could rewrite the premise (which is $ A \wedge B $) but I didn't figured it out.

Answer 1

From $A\land B$ you can infer $A$, and from $A$ you can infer $A\lor B$, so from $A\land B$ you can infer $A\lor B$.

Another way to think about it. The following are all tautologies, hence provable: $$\begin{align} &\vdash (A\land B)\to A \\ &\vdash A\to(A\lor B) \\ &\vdash (p\to q)\to((q\to r)\to(p\to r)) \tag{all instances are provable} \end{align}$$

From these you can derive $(A\land B)\to (A\lor B)$.