Maths WorkshopMaths problems with full solutions

Proof techniques

44 problems · Direct proof, proof by contradiction, contraposition and cases.

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Ex.248Proof by counterexample

#counterexample #disproof #universal statement

Proof techniques

Ex.249De Morgan's laws

#De Morgan #logic #negation

Proof techniques

Ex.250Direct proof

#direct proof #implication #even odd

Proof techniques

Ex.251Geometric sum by induction

#induction #geometric series #sum

Ex.252Proof by induction: sum of first $n$ integers

#induction #sum #integers

Ex.253Summary of induction

#summary #induction #techniques

Ex.323Bernoulli's inequality

#induction #inequalities

Ex.337Sum of squares by induction

#induction #sums

Ex.495Proof by contradiction

#contradiction #irrationality #sqrt2

Proof techniques

Ex.496Proof by contraposition

#contrapositive #implication #odd

Proof techniques

Ex.497Contrapositive with primes

#contrapositive #prime #divisibility

Proof techniques

Ex.498Converse, inverse, contrapositive

#implication #converse #contrapositive

Proof techniques

Ex.499Bernoulli's inequality by induction

#induction #Bernoulli #inequality

Ex.500Sum of squares by induction

#induction #sum of squares #integers

Ex.501Proof by cases

#proof by cases #parity #absolute value

Proof techniques

Ex.502A parity case analysis

#cases #parity #square

Proof techniques

Ex.503A tautology

#propositional logic #tautology #truth table

Proof techniques

Ex.504Negating quantifiers

#quantifiers #negation #logic

Proof techniques

Ex.505Summary of logic and proof techniques

#summary #proof #logic

Proof techniques

Ex.1031Proving an equivalence

#equivalence #iff #proof

Proof techniques

Ex.1032Coins and strong induction

#induction #coins #Frobenius

Ex.1033Full binary tree node count

#induction #binary tree #structural

Ex.1034Divisibility by induction

#induction #divisibility #powers

Ex.1035An identity for Fibonacci numbers

#induction #Fibonacci #identity

Ex.1036Towers of Hanoi

#induction #Hanoi #recurrence

Ex.1037Exponential inequality by induction

#induction #inequality #exponential

Ex.1038Fibonacci growth bound

#induction #Fibonacci #growth

Ex.1039$n!$ vs $2^n$

#induction #factorial #exponential

Ex.1040Induction and periodicity

#induction #periodicity #recurrence

Ex.1041Product by induction

#induction #product #telescoping

Ex.1042Strong induction

#strong induction #prime factorization

Ex.1043Sum of cubes

#induction #sum of cubes

Ex.1044A factorial identity

#induction #factorial #identity

Ex.1045Existence proofs

#existence #constructive #non-constructive

Proof techniques

Ex.1046Order of quantifiers

#quantifiers #order #logic

Proof techniques

Ex.1283Two implications imply the negation of $p$

#implication #truth table #contradiction

Proof techniques

Ex.1755Euler characteristic by induction

#induction #Euler characteristic #trees

Ex.1756Determinant of a matrix by induction

#induction #matrix #determinant

Ex.1757Existence of quotient and remainder

#strong induction #division #quotient

Ex.1758Every positive integer is a sum of non-consecutive Fibonacci numbers

#induction #Fibonacci #Zeckendorf

Ex.1759Representation of integers

#strong induction #representation #powers

Ex.1760Tiling by induction

#induction #tiling #L-tromino

Ex.1761Russell's paradox (statement)

#Russell paradox #sets #self-reference

Proof techniques

Ex.2177Real induction (statement)

#real induction #intervals #completeness