Conjunction and Disjunction

Mathematical Logic · Axiom Academy

LESSON Conjunction and Disjunction Understanding AND and OR operations in propositional logic 1. Conjunction (∧): The AND Operation The conjunction operator represents the logical "and". Think of it as a strict requirement: both conditions must be satisfied simultaneously. "It is raining and it is cold" - True only when both conditions hold "I am a student and I am hungry" - Both must be true "The light is on but nobody is home" - "But" is also conjunction 2. Disjunction (∨): The OR Operation The disjunction operator represents the inclusive OR . Unlike everyday language where "or" sometimes means "one or the other but not both," in logic ∨ includes the case where both are true. "I will study math or I will study physics" - True if I study either or both "You can have tea or coffee" - Typically exclusive in practice, but logically inclusive "It is weekend or it is a holiday" - True on days that are either or both 3. Exclusive OR (⊕): The XOR Operation While not one of the fundamental connectives, XOR is important in many contexts. It captures the "either...or" meaning often intended in natural language. "You will pass or you will fail" - Exactly one must be true "Either we go left or we go right" - Cannot do both simultaneously "The light is on xor the light is off" - Mutually exclusive states Both conjunction and disjunction satisfy important algebraic properties that allow us to manipulate logical expressions: Order doesn't matter for AND or OR operations.

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