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As the OP and the comments note, both of the results are tautologies. Here is the truth table for the first one: Here is the truth table for the second one: As the OP notes the two sides of the biconditional are also tautologies. As to why these were marked false, perhaps the answer key was in error or the problem that was intended was misstated. Perhaps ...


You could rewrite "A or B" as "(A and not B) or (not A and B) or (A and B)". You can see that these statements are equivalent using a truth table generator. Using the input "((A&&~B)or(~A&&B)or(A&&B))<=>(A or B)" in this generator I find that all valuations of "A" and "B" give "T" values. You could then decide with a separate if-...

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