# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) [← Previous](/en/derivations/?page=1#results)[1](/en/derivations/?page=1#results) 2 [3](/en/derivations/?page=3#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=3#results) 299 results · Page 2 of 13 Concept guide ## [Basic counting principles: how many three-digit even numbers have distinct digits](/en/derivations/derivation-mko-counting-principle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Count the even integers from 100 to 999 whose three digits are pairwise distinct. Requirements: add only disjoint classes, and before each multiplication confirm that the number of options at every stage is the same for all earlier choices; when it is not, branch first. Basic counting principles derivation-mko-counting-principle-01 Concept guide ## [Factorial: uniqueness of the recursive definition, the product form and computing 5!](/en/derivations/derivation-mko-factorial-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Using only F(0)=1 and F(n+1)=(n+1)F(n) (n∈ℕ, with 0∈ℕ): (1) prove that at most one function ℕ→ℕ satisfies these two equations; (2) prove F(n)=∏_{k=1}^{n} k for every n (the empty product being 1), so that for n>0, n! is the product of all integers from 1 to n; (3) prove F(n)≥1 for every n, in particular 0!=1≠0; (4) compute 5!=120 step by step from the recursion. Factorial derivation-mko-factorial-01 Concept guide ## [Absolute value: nonnegativity, the zero case and |a−b|=|b−a| from the piecewise definition](/en/derivations/derivation-mko-absolute-value-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Using only the piecewise definition (|x|=x when x≥0, |x|=−x when x<0) together with the order and addition of the reals, prove for all reals x, a, b: (1) |x|≥0; (2) |x|=0 ⇔ x=0; (3) |−x|=|x|; (4) |a−b|=|b−a|. Then compute |−3|, |2−7| and |7−2|, and show that '|x|=x for every x' is false. Absolute value derivation-mko-absolute-value-01 Concept guide ## [Fractions: deciding equal value, normalizing to lowest terms with a positive denominator, and adding fractions](/en/derivations/derivation-mko-fraction-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) (1) Decide whether (−6)/(−8) and 9/12 represent the same rational number; (2) bring (−6)/(−8) to lowest terms with a positive denominator and show that the value is unchanged; (3) compute 3/4+5/6 and show that 'numerator plus numerator over denominator plus denominator', giving 8/10, is not the sum; (4) explain why 3/0 is not a valid fraction representation. Fraction derivation-mko-fraction-01 Concept guide ## [Ratio and proportion: solving 3:4=x:10, checking back, and testing data for direct proportion](/en/derivations/derivation-mko-ratio-proportion-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Over the rationals: (1) prove that for b≠0 and d≠0 the proportion a:b=x:d has exactly one rational solution, x=ad/b; (2) use it to solve 3:4=x:10 and check back; (3) show that 4:3=x:10 has a different solution, so the order of a ratio matters; (4) decide whether the data (2,5), (4,10), (6,15), (8,21) fit a single direct proportion y=kx. Ratio and proportion derivation-mko-ratio-proportion-01 Concept guide ## [Domain, codomain and range: a finite table and the range of h(x)=x²+1](/en/derivations/derivation-mko-domain-codomain-range-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) (1) With A={−1,0,1,2}, B={0,1,2,3,4,5} and f(x)=x²+1: verify that f:A→B is a total function, find the range f[A], and show f[A]⊆B but f[A]≠B; (2) explain why data with an output outside the declared codomain, or with a missing input, is not a total function in the sense of this entry; (3) for h:ℝ→ℝ, h(x)=x²+1, prove that the range is [1,∞), which differs from the codomain ℝ. Domain, codomain, and range derivation-mko-domain-codomain-range-01 Concept guide ## [Function composition: the permitted inputs of a restricted composition, evaluation order and non-commutativity](/en/derivations/derivation-mko-composition-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let f:ℝ→ℝ, f(x)=x−3, and g:[0,∞)→ℝ, g(u)=√u. (1) Find the permitted inputs D={x∈dom(f) ∣ f(x)∈dom(g)} of g∘f and prove D=[3,∞); (2) evaluate (g∘f)(7) and (g∘f)(4) by 'f first, then g', and explain why (g∘f)(1) is undefined; (3) show that f∘g is a total composition on [0,∞), and conclude g∘f≠f∘g from (f∘g)(4)=−1≠1=(g∘f)(4); (4) show that composition is not multiplication of function values. Function composition derivation-mko-composition-01 Concept guide ## [Sequences: fixing the general term from an initial condition and a recurrence, and the pitfalls of indexing and finite prefixes](/en/derivations/derivation-mko-sequence-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let the real sequence a:ℕ→ℝ (with 0∈ℕ) satisfy a₀=5 and a_{n+1}=a_n+3. (1) Prove that exactly one such sequence exists and that a_n=5+3n; (2) compute a₀ to a₄ and a₁₀, explaining that a₀ is the initial term and a₁₀ is the 11th value when counting from 1; (3) write down the finite sequence of length L=4 and its index set; (4) show that the prefix 5, 8, 11 does not determine the sequence. Sequence derivation-mko-sequence-01 Concept guide ## [Permutations and combinations: proving P(n,k)=n!/(n−k)! and C(n,k)=n!/(k!(n−k)!) by counting](/en/derivations/derivation-mko-permutation-combination-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let X be a set of exactly n distinct objects, with natural numbers n, k and 0≤k≤n, and selection without replacement and without repetition. Prove that the number of ordered selections of k objects from X is P(n,k)=n!/(n−k)!, that the number of unordered selections is C(n,k)=n!/(k!(n−k)!), and that both equal 1 when k=0. Finally compute P(5,3)=60 and C(5,3)=10 for X={a,b,c,d,e} and k=3. Permutations and combinations derivation-mko-permutation-combination-01 Concept guide ## [Coordinate plane: deciding axis or quadrant membership and comparing ordered pairs](/en/derivations/derivation-mko-coordinate-plane-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) In a fixed Cartesian coordinate system, let A=(−3,2), B=(2,−3), B′=(6/3,−3), C=(0,−4), O=(0,0) and E=(−1/2,−5). First prove that the axes meet only at the origin and that every point either lies on an axis or lies in exactly one quadrant; then decide for each point its axis or quadrant, and decide whether A and B, and B and B′, are the same point. Coordinate plane derivation-mko-coordinate-plane-01 Concept guide ## [Natural-number powers and principal square roots: computing powers by recursion and proving √(a²)=|a|](/en/derivations/derivation-mko-power-root-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Using only the recursive definition a^0=1, a^(n+1)=a^n·a, compute (−2)^n for n=0,1,2,3,4 and 0^0, 0^3; prove that a nonnegative square root is unique and that √(a²)=|a| for every real a; then find √49, √((−5)²) and the real solution set of y²=9, and explain why √(−4) is not defined in the reals. Natural-number powers and principal square roots derivation-mko-power-root-01 Concept guide ## [Univariate rational polynomials: combining like terms, coefficient vectors, degree and the zero polynomial](/en/derivations/derivation-mko-polynomial-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) In ℚ[x], let p(x)=4x³+3x²−x+2−4x³+x, let q have coefficient vector [−1,1,0,0], and let r(x)=x²−x²+3, z(x)=5x−5x and t(x)=5x. Find the coefficients and degrees of p, q and r; show that z is the zero polynomial and distinguish it from r; compute p+q and p·q from coefficients and find their degrees; finally explain why p(1)=t(1) does not give p=t. Univariate polynomials over the rationals derivation-mko-polynomial-01 Concept guide ## [Polynomial factorization over the rationals: verifying common factors, binomial products and differences of squares by comparing coefficients](/en/derivations/derivation-mko-factorization-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) In ℚ[x]: (1) factor p(x)=3x²+3x−18 as 3(x+3)(x−2); (2) factor q(x)=2x²−50 as 2(x−5)(x+5); verify each factorization by expanding and comparing every coefficient. (3) Prove that x²+25≠(x−5)(x+5) and that x²+25 is not a product of two degree-one factors. (4) Explain why x³ and x take equal values at x=−1, 0, 1 and still are different polynomials. Polynomial factorization over the rationals derivation-mko-factorization-01 Concept guide ## [Affine and degree-one functions: finding m and b from two points, and deciding degree, zero and linearity](/en/derivations/derivation-mko-linear-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) The graph of the affine function f:ℝ→ℝ, f(x)=mx+b, passes through (1,4) and (3,0). Find m and b; decide whether f has degree one, find its zero, and decide whether f is an ℝ-linear map in the sense of linear algebra. Then prove that two affine functions are equal on ℝ exactly when their slopes and intercepts agree, and use it to explain why f(2)=k(2) (with k(x)=x) does not give f=k; finally treat the two degenerate examples with m=0, g(x)=5 and h(x)=0. Affine functions and degree-one functions derivation-mko-linear-function-01 Concept guide ## [Graphs of real-valued functions: membership tests and verifying a finite table as a total function](/en/derivations/derivation-mko-function-graph-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) (1) Let D=[0,2], f:D→ℝ, f(x)=x²−1. First prove (u,v)∈G_f ⇔ u∈D and v=f(u), then decide whether (1,0), (−1,0), (2,2) and (2,3) lie on G_f. (2) Let D_0={0,1,2} and B={0,1,4}. Prove that a set Γ is the graph of some total function D_0→B exactly when Γ⊆D_0×B and every input has exactly one output; use it to decide that T={(0,0),(1,1),(2,4)} is such a graph, while T_1={(0,0),(2,4)}, T_2={(0,0),(1,1),(1,4),(2,4)} and T_3={(0,0),(1,1),(2,4),(3,9)} are not. Graphs of real-valued functions derivation-mko-function-graph-01 Concept guide ## [Probability on finite sample spaces: validating a weight model, computing event probabilities and complements](/en/derivations/derivation-mko-probability-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let S={1,2,3,4} with the full powerset as event family and weights w_1=1/2, w_2=1/3, w_3=1/6, w_4=0. Verify that this is a legal finite probability model; prove from the definition the identities P(∅)=0 and P(S)=1, additivity for disjoint events, and the complement identity P(S∖E)=1−P(E); compute P({1,3}) and explain why it differs from |E|/|S|; show that {4} is nonempty yet has probability 0; and decide that the candidate weights (1/2,1/3,1/3,0) and (1/2,2/3,−1/6,0) are both illegal. Probability on finite sample spaces derivation-mko-probability-01 Concept guide ## [Quantifiers and explicit domains: deciding universal and existential statements and their negations on a finite integer domain](/en/derivations/derivation-mko-quantifier-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) On the explicit finite domain U={−2,−1,0,1,2,3}, decide the truth of φ₁: ∀x∈U, x²<10; φ₂: ∀x∈U, x²>x; φ₃: ∃x∈U, x²=2; write the negations of φ₂ and φ₃ correctly; and settle the truth of universal and existential statements when the domain is the empty set. Quantifiers and explicit domains derivation-mko-quantifier-01 Concept guide ## [Counterexamples and domain eligibility: primality of n²+n+11, the failure set and the least counterexample](/en/derivations/derivation-mko-counterexample-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let U={n∈ℤ ∣ 0≤n≤12}, V={n∈ℤ ∣ 0≤n≤9}, m(n)=n²+n+11 and P(n): 'm(n) is prime'. Decide φ_U: ∀n∈U P(n); find the failure set F_U and the least counterexample; decide whether the candidate −11 is eligible; and decide φ_V: ∀n∈V P(n). Counterexamples and domain eligibility derivation-mko-counterexample-01 Concept guide ## [Binary relations: computing the domain, images and the converse step by step](/en/derivations/derivation-mko-relation-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let A={1,2,3,4}, B={2,3,4,5,6}, R={(1,2),(1,3),(3,3),(4,6)} and C={1,3}. Check R⊆A×B; compute Dom(R), R[C], R[∅], R⁻¹ and Dom(R⁻¹); decide whether R is a function A→B and whether R⁻¹ is a function B→A. Binary relations and set projections derivation-mko-relation-01 Concept guide ## [Finite simple undirected graphs: neighbours, degrees and the handshake formula](/en/derivations/derivation-mko-graph-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let V={1,2,3,4,5}, with edges entered as a list of endpoint pairs: (1,2), (2,1), (2,3), (3,4), (2,4), (4,2), each entry denoting an undirected edge. Normalize the input to the edge set E, find the neighbour set N(v) and degree deg(v) of every vertex, check that the degree sum equals 2|E|, and prove the handshake formula Σ_{v∈V} deg(v)=2|E| for every finite simple undirected graph. Finite simple undirected graphs derivation-mko-graph-01 Concept guide ## [Plane vectors in a fixed basis: componentwise operations, equality and the direction of subtraction](/en/derivations/derivation-mko-vector-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) In the fixed standard basis let u=(3,−1), v=(−2,4), w=(1,3) and k=−2. Compute u+v, u−v, v−u, ku, 0u and 2u+v componentwise; find the vector x with u+x=v; check k(u+v)=ku+kv; and decide whether u=w and whether u=(−1,3). Plane vectors in a fixed basis derivation-mko-vector-01 Concept guide ## [Linear systems in two variables: the unique-solution formula, checking both rows, and consistency when D=0](/en/derivations/derivation-mko-system-linear-equations-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) (I) Solve 2x+3y=7, x−y=1: prove the unique-solution formula when D=ae−bd≠0, solve, and check by substituting into both rows; (II) decide x+2y=3, 2x+4y=7; (III) decide x+2y=3, 2x+4y=6; and use (IV) 0x+0y=1, 0x+0y=2 to show that D and both Cramer numerators being zero does not imply infinitely many solutions. Linear systems in two variables and degenerate constraints derivation-mko-system-linear-equations-01 Concept guide ## [Conditional probability in finite models: positive denominators, the two directions and null conditioning events](/en/derivations/derivation-mko-conditional-probability-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) In the finite model on S={1,2,3,4,5,6} with weights w=(1,2,3,2,4,0)/12, let A={2,3}, B={3,4,5,6} and Z={6}. Validate the model, compute P(A|B) and P(B|A), compare them with the equiprobable count ratio |A∩B|/|B| and with the product P(A)P(B), and decide whether P(A|Z) is defined. Conditional probability in finite models derivation-mko-conditional-probability-01 Concept guide ## [Injectivity and distinct inputs: table qualification, collision witnesses and fibers](/en/derivations/derivation-mko-injective-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D2) Let A={1,2,3,4}, B={1,2,3,4,5,6}, with function tables f: (1,4),(2,6),(3,1),(4,5); g: (1,4),(2,6),(2,6),(3,4),(4,6); h: (1,4),(2,6),(2,5),(3,1),(4,5). Decide which tables are total functions A→B; decide whether f and g are injective and list their fibers; find g's first collision in lexicographic order; decide whether g restricted to C={1,2} is injective; and treat the empty domain. Injectivity and distinct inputs derivation-mko-injective-01 [← Previous](/en/derivations/?page=1#results)[1](/en/derivations/?page=1#results) 2 [3](/en/derivations/?page=3#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=3#results) 299 results · Page 2 of 13