Answer to Question #160928 in Discrete Mathematics for Bob Manley

Question #160928

I am having trouble with the following problem. I am unsure on how to even start.


Use propositional logic to prove that the following argument are valid:

(A⟶B) ⋀ (A⟶(B⟶C)) ⟶ (A⟶C)



1
Expert's answer
2021-02-12T17:33:56-0500

"(A\\to B) \\wedge (A\\to(B\\to C)) \\to(A\\to C) \\\\\n(A\\to B) \\wedge (A\\to(B\\to C))\\\\\n(A'\\vee B) \\wedge(A \\to(B' \\vee C)) \\text{Implication} \\\\\n(A'\\vee B) \\wedge(A' \\vee (B'\\vee C)) \\text{Implication}\\\\\n(A'\\vee B) \\wedge (A'\\vee B') \\vee C \\text{Assoiativity}\\\\\nA'\\vee(B \\wedge B') \\vee C \\text{Distributive}\\\\\n(A'\\vee 0)\\vee C \\text{Known Contradiction}\\\\\nA'\\vee C \\text{Absorbtion}\\\\\n(A\\to C)"

This shows that "(A\\to B) \\wedge (A\\to(B\\to C)) \\to(A\\to C)". Hence the argument is valid.


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