Negating quantifications

• Consider the statement: – All students in this class have red hair Predicates and Quantifiers • What is required to show the statement is false? – There exists a student in this class that does NOT have red hair CS 231, Fall 2008 • To negate a universal quantification: Dianna Xu – Negate the predicate – AND change to an existential quantification – Change “All are” to “Some are not” – ~(∀x in D, P(x)) ≡ ∃x in D, ~P(x)

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Negating quantifications 2 More negations

• Consider the statement: • “No man is an island” – John Donne – There is a student in this class with red hair – ∀x, x is not an island • What is required to show the statement is – ∃x, x is an island false? – Some men are islands – All students in this class do not have red hair • “All that glitters is not gold” – Chaucer and • Thus, to negate an existential quantification: Shakespeare – Negate the predicate – AND change to a universal quantification – ~(∃x in D, P(x)) ≡ ∀x in D, ~P(x) 3 4

Negation of universal ∀ and ∧ conditionals • Given a predicate P(x) and values in the • ~(∀x in D, P(x) → Q(x)) ≡ ∃x in D, ~(P(x) domain {x , … , x } → Q(x)) 1 n • ~(∀x in D, P(x) → Q(x)) ≡ ∃x in D, P(x) ∧ ~Q(x) • The universal quantification ∀x P(x) • For all people x, if x is rich then x is happy implies: • There is one person who is rich and is not happy • There is one person who is rich but not happy P(x1) ∧ P(x2) ∧ … ∧ P(xn)

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1 ∃ and ∨ Translating from English

• Given a predicate P(x) and values in the • Translate the statements: domain {x , …, x } – “All hummingbirds are richly colored” 1 n – “No large birds live on honey” – “Birds that do not live on honey are dull in color” • The existential quantification ∃x P(x) – “Hummingbirds are small” implies: • Assign our predicates – Let P(x) be “x is a hummingbird” – Let Q(x) be “x is large” – Let R(x) be “x lives on honey” P(x1) ∨ P(x2) ∨ … ∨ P(xn) – Let S(x) be “x is richly colored” • Let our domain be all birds 7 8

Translating from English

• “All hummingbirds are richly colored” – ∀x, (P(x)→S(x)) • “No large birds live on honey” – ~∃x, (Q(x) ∧ R(x)) – Alternatively: ∀x, (~Q(x) ∨ ~R(x)) • “Birds that do not live on honey are dull in color” – ∀x, (~R(x) → ~S(x)) • “Hummingbirds are small”

– ∀x, (P(x) → ~Q(x)) 9

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