Worked examples

Sets, induction and elementary functions

One sheet for each type of problem, with several worked examples of increasing difficulty: first the computation, then the explanation in words. Each sheet starts with links to the theory it uses.

• 1Operations on given sets
• 2Supremum and infimum of number sets
• 3Verifying sup and inf with the epsilon characterization
• 4Proofs of irrationality by contradiction
• 5Proofs by induction of summation formulas
• 6Proofs by induction of inequalities
• 7Proofs by induction of divisibility properties
• 8Computations with binomial coefficients and the binomial theorem
• 9Domains of composite functions
• 10Injectivity and inverse functions
• 11Parity and symmetry
• 12Graphs obtained by transformations
• 13Exponential inequalities
• 14Logarithmic inequalities
• 15Trigonometric equations
• 16Ranges of functions