I am wondering how to prove if A, then B or C. I saw one proof is to show if A is true and B is false, then C is true. I think it’s true but can’t reason why. Could someone show me the logic behind it? Also wonder are there other ways to prove it, it seems a very common type of proof.
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1$p \to q$ is the same as $\neg p \vee q$, so, $B \vee C$ is the same as $\neg B \to C$. – azif00 Aug 15 '20 at 01:42
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We want to prove: "If $A$, then $B$ or $C$."
Consider the two cases:
- $B$ is true.
If $A$ is true and $B$ is true then we are done, since $B$ or $C$ is true.
- $B$ is false.
If $A$ is true and $B$ is false, if we manage to prove that $C$ is true in this setting, then $B$ or $C$ is evaluated to be true as well.
We have covered every single case.
Siong Thye Goh
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Thank you! Then I understand why some say that one only needs to prove if A, not B, then C. – Coco Aug 22 '20 at 00:44