Maths WorkshopMaths problems with full solutions

Quantifiers and negation

37 problems · Universal and existential quantifiers, predicate logic and logical equivalences.

Ex.267The existential quantifier
#existential quantifier #exists #example
Quantifiers and negation
High school
High school
Ex.268The universal quantifier
#universal quantifier #forall #proposition
Quantifiers and negation
High school
High school
Ex.269Double negation
#double negation #logic #equivalence
Quantifiers and negation
High school
High school
Ex.270Proving an existential statement
#existential proof #example #witness
Quantifiers and negation
High school
High school
Ex.271Universal over a finite set
#finite quantifier #forall #conjunction
Quantifiers and negation
High school
High school
Ex.329Negating convergence
#quantifiers #sequences
Quantifiers and negation
L1
L1
Ex.558Bounded quantifiers
#bounded quantifiers #notation #domain
Quantifiers and negation
L1
L1
Ex.559Quantifier over a finite set
#finite quantifier #exists #disjunction
Quantifiers and negation
High school
High school
Ex.560Negating an existential statement
#negation #existential #exists
Quantifiers and negation
High school
High school
Ex.561Negating a universal statement
#negation #universal #forall
Quantifiers and negation
High school
High school
Ex.562Summary of quantifiers and negation
#summary #quantifiers #negation
Quantifiers and negation
High school
High school
Ex.563The uniqueness quantifier
#uniqueness #exists unique #quantifier
Quantifiers and negation
L1
L1
Ex.564Refuting a universal statement
#refutation #counterexample #forall
Quantifiers and negation
High school
High school
Ex.1150Quantifier absorption
#quantifier absorption #vacuous quantifier
Quantifiers and negation
L1
L1
Ex.1151Renaming bound variables
#alpha-conversion #bound variable #renaming
Quantifiers and negation
L1
L1
Ex.1152De Morgan's laws for quantifiers
#De Morgan #quantifiers #negation
Quantifiers and negation
L1
L1
Ex.1153The $\varepsilon$-$\delta$ definition of limit
#epsilon-delta #quantifiers #limit
Quantifiers and negation
L1
L1
Ex.1154Existence and uniqueness proof
#existence #uniqueness #proof
Quantifiers and negation
L1
L1
Ex.1155A false $\exists\forall$ statement
#exists forall #false #counterexample
Quantifiers and negation
L1
L1
Ex.1156A quantified implication
#forall implies #existential #mixed
Quantifiers and negation
L1
L1
Ex.1157Free and bound variables
#free variable #bound variable #quantifier
Quantifiers and negation
L1
L1
Ex.1158Quantifiers in induction proofs
#induction #quantifiers #base case
Quantifiers and negation
L1
L1
Ex.1159Quantifiers in convergence of sequences
#convergence #epsilon #sequences
Quantifiers and negation
L1
L1
Ex.1160Negating nested quantifiers
#negation #nested quantifiers #forall exists
Quantifiers and negation
L1
L1
Ex.1161Negating the uniqueness quantifier
#negation #uniqueness #exists unique
Quantifiers and negation
L1
L1
Ex.1162Order of quantifiers matters
#quantifier order #forall exists #difference
Quantifiers and negation
L1
L1
Ex.1163Vacuous truth
#vacuous truth #forall #empty set
Quantifiers and negation
L1
L1
Ex.1284The barber's paradox
#self-reference #paradox #implication
Quantifiers and negation
L1
L1
Ex.1285Three numbers with signs
#sign analysis #implications #logic puzzle
Quantifiers and negation
L1
L1
Ex.1857Alternation depth of quantifiers
#quantifier alternation #depth #complexity
Quantifiers and negation
L1
L1
Ex.1858Scope of quantifiers
#scope #quantifiers #ambiguity
Quantifiers and negation
L1
L1
Ex.1859When distribution fails
#distribution #counterexample #quantifier
Quantifiers and negation
L1
L1
Ex.1860Proving a quantified statement with $\exists\forall$
#exists forall #nested #olympiad
Quantifiers and negation
Olympiad
Olympiad
Ex.1861Logical equivalence of quantified statements
#equivalence #quantifiers #distribution
Quantifiers and negation
L1
L1
Ex.1862Negating a complex statement
#negation #practice #nested
Quantifiers and negation
L1
L1
Ex.1863Prenex normal form
#prenex form #quantifiers #normal form
Quantifiers and negation
L1
L1
Ex.1864A statement with a specific structure
#epsilon #forall #exists
Quantifiers and negation
L1
L1