Negation and contrapositive

Japanese school year: Math I

What you learn

You will learn how to formulate negations, converses, inverses, and contrapositives of propositions, exploring why a statement shares its truth value with its contrapositive. This concept is crucial for proving mathematical statements indirectly. Understanding basic set relationships and deductive reasoning will support your progress.

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Key points

This rule states that " $p$ implies $q$ " and its contrapositive "not $q$ implies not $p$ " always have the exact same truth value. Use it to prove a statement indirectly when the direct claim is hard to prove.

(p \implies q) \iff (\neg q \implies \neg p)

Negating " $p$ and $q$ " is logically equivalent to "not $p$ , or not $q$ " (De Morgan's law). Use this rule when you want to negate two conditions holding at the same time.

\neg (p \land q) \iff \neg p \lor \neg q

Negating " $P(x)$ holds for all $x$ " means "there is at least one $x$ where $P(x)$ does not hold." Use this to disprove a universal claim by showing a single counterexample.

\neg (\forall x, P(x)) \iff \exists x, \neg P(x)

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Practice 1Coming soon