Statements and quantifiers
Objective. Negate and combine quantified mathematical statements.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A mathematical statement is a sentence that is unambiguously true or false, and quantifiers (∀ "for all", ∃ "there exists") specify how many objects satisfy a predicate P(x). To negate ∀x P(x) you get ∃x ¬P(x), and to negate ∃x P(x) you get ∀x ¬P(x); negation always flips the quantifier and negates the inner statement. For compound statements, ¬(A∧B) becomes ¬A∨¬B and ¬(A∨B) becomes ¬A∧¬B by De Morgan's laws. For example, the negation of "∀n∈ℕ, n²≥0" is "∃n∈ℕ, n²<0", which is false, confirming the original claim is true. Mastering this swap is essential before writing any proof that proceeds by contradiction.
Worked example
Write the negation of the statement "∀x∈ℝ, x>0 ⇒ x²>0" and state whether the original statement is true.
- 1Flip ∀ to ∃: the negation begins "∃x∈ℝ".
- 2Negate the implication P⇒Q as P∧¬Q, giving x>0 ∧ x²≤0.
- 3So the negation is "∃x∈ℝ, x>0 ∧ x²≤0".
- 4Check: no positive real has x²≤0, so the negation is false and the original statement is true.
Answer: ∃x∈ℝ, x>0 ∧ x²≤0 (negation); original statement is true
Concept mastery
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Practice
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Start practisingCommon mistake. Students often negate "∀x, P(x)⇒Q(x)" as "∀x, P(x)⇒¬Q(x)" instead of "∃x, P(x)∧¬Q(x)"; negating an implication needs AND, not another implication. Always negate the whole quantified implication as one block.
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