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How to | Use Logical Operators

The Wolfram Language supports logical operators not only for programming, but for mathematical operations as well.

The infix operators && and || stand for conjunction (And) and disjunction (Or), while ! is the prefix operator for negation (Not). The next two inputs are equivalent:

Wolfram Language code: (True && False) || !True
Wolfram Language code: Or[And[True, False], Not[True]]

You can use symbols instead of True and False. BooleanConvert symbolically evaluates many logical expressions. We expect to be :

Wolfram Language code: BooleanConvert[(a && b) || (!a && b)]

Use Resolve to find truth values of quantified logical statements:

Wolfram Language code: Resolve[ForAll[{a, b}, Equivalent[(a && b) || (!a && b), b]], {a, b}, Booleans]

You can also use the existential quantifier to check that :

Wolfram Language code: Resolve[ForAll[a, Exists[b, (a && b) || (!a && b)]], {a, b}, Booleans]

The Wolfram Language also recognizes the logical operator Implies; you can use it to show :

Wolfram Language code: Resolve[ForAll[{a, b}, Implies[a && b, b]], {a, b}, Booleans]
    

Use SatisfiableQ to check if a and b satisfy :

Wolfram Language code: SatisfiableQ[(a && b) && (!a || !b), {a, b}]