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In the literal sense, asserting that A "implies" B (or A ⇒ B) means that if A is true, B must also be true. It says nothing about B when A is false. Therefore: When A is true, B must also be true (because of the implication itself). When A is false, we don't know anything about B, so B could be either true or false. The binary operation "A implies B" ...


FullSimplify[Not[a] || (a && b)] // InputForm Implies[a, b] Simplify[Not[a] || (a && b)] ! a || b BooleanConvert[Implies[a, b]] ! a || b TableForm[BooleanTable[{a, b, ! a || b}, {a, b}], TableHeadings -> {None, {a, b, ! a || b}}]


The logical simplification of Not[a] || (a && b) is indeed Implies[a,b]. You can see this by comparing the logical truth tables: Table[{a, b, Not[a] || (a && b)}, {a, {False, True}}, {b, {False, True}}] // MatrixForm is the same as Table[{a, b, Implies[a, b]}, {a, {False, True}}, {b, {False, True}}] // MatrixForm ...

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