# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) 1 [2](/en/derivations/?page=2#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=2#results) 299 results · Page 1 of 13 Concept guide ## [Points, lines and planes: deciding incidence in the coordinate model](/en/derivations/derivation-mko-point-line-plane-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the coordinate-plane model ℝ², let P=(1,2) and Q=(3,6). Decide whether R=(4,8) and S=(2,5) lie on the line L through P and Q, and hence whether P, Q, R are collinear. Points, lines, and planes derivation-mko-point-line-plane-01 Concept guide ## [Angle: a step-by-step proof that vertical angles are equal](/en/derivations/derivation-mko-angle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Distinct lines AB and CD meet at O, with O between A and B and between C and D. Prove that vertical angles are equal: m∠AOC=m∠BOD and m∠COB=m∠DOA; then, for m∠AOC=35°, compute all four angles. Angle derivation-mko-angle-01 Concept guide ## [Triangle: confirming in the coordinate model that three points determine a nondegenerate triangle](/en/derivations/derivation-mko-triangle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the coordinate-plane model ℝ², let A=(0,0), B=(4,0), C=(1,3). Confirm that A, B, C are pairwise distinct and noncollinear, so that by the MKO they determine the nondegenerate triangle ABC; write out its three sides and the side opposite each vertex, and show that the average of the vertices, G=(5/3,1), is not on triangle ABC. Then, with D=(2,1) and E=(6,3), show that the distinct points A, D, E are collinear and so do not determine a nondegenerate triangle. Triangle derivation-mko-triangle-01 Concept guide ## [Right triangle: deciding in coordinates which interior angle is right](/en/derivations/derivation-mko-right-triangle-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the coordinate-plane model ℝ², let A=(1,1), B=(5,3), C=(0,3). Prove that A, B, C form a nondegenerate triangle whose interior angle at A equals 90° (that is, π/2), so ABC is a right triangle; show that the other two angles are not right and that any nondegenerate triangle has at most one right angle, and identify the hypotenuse and the legs. Then, with D=(1,4), show that triangle ABD is not a right triangle. Right triangle derivation-mko-right-triangle-01 Concept guide ## [Euclidean length: computing the distance between two points exactly from coordinates](/en/derivations/derivation-mko-euclidean-length-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the Euclidean plane represented by ℝ², let P=(1,−2), Q=(4,2), R=(2,3). Compute d(P,Q)=5 exactly; compute d(P,R)=√26 and prove by comparing squares that 5.09<√26<5.1, showing that 5.1 is only an approximation. For general real coordinates, also prove that the distance is defined and nonnegative, that d(Q,P)=d(P,Q), that d(P,P)=0, and that d(P,Q)=0 ⇔ P=Q. Euclidean length derivation-mko-euclidean-length-01 Concept guide ## [Pythagorean theorem: a proof by dot products in the coordinate model](/en/derivations/derivation-mko-euclid-pythagorean-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the Euclidean plane represented by ℝ² (with Euclidean length as distance), prove: for every right triangle ABC with the right angle at C, legs of lengths a=d(C,B) and b=d(C,A), and hypotenuse of length c=d(B,A), a²+b²=c². Then check the example C=(1,2), B=(4,6), A=(−7,8) (25+100=125), and use the non-right triangle E=(0,0), F=(4,0), H=(1,2) to show that without a right angle the relation fails in general. Pythagorean theorem derivation-mko-euclid-pythagorean-theorem-01 Concept guide ## [Natural number: deciding membership through the smallest inductive set, and telling numbers from numerals](/en/derivations/derivation-mko-natural-number-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Under the OCME convention (0∈N), read the MKO's '0 and the numbers obtained from 0 by finitely many successor steps' as 'the smallest set of reals that contains 0 and is closed under x↦x+1'. On this basis prove 0∈N and 3∈N; show that the binary numeral '11' and the decimal numeral '3' denote the same natural number; prove −1∉N and 1/2∉N; and compare least elements with the convention N⁺=N∖{0}, in which natural numbers start at 1. Natural number derivation-mko-natural-number-01 Concept guide ## [Integer: deciding membership, the meaning of the minus sign, and closure under negation](/en/derivations/derivation-mko-integer-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Inside the reals, take Z={0}∪N⁺∪{−n ∣ n∈N⁺} (N contains 0 by the OCME convention, N⁺=N∖{0}, and −n is the additive inverse of n). Prove −7∈Z; show that the decimal numeral '3.0' denotes the integer 3; prove −2.5∉Z; prove −(−a)=a from uniqueness of additive inverses, in particular −(−4)=4; prove −z∈Z for every z∈Z; and use 5−8=5+(−8)=−3 to tell binary subtraction from the unary minus sign. Integer derivation-mko-integer-01 Concept guide ## [Rational number: different fraction representations of one rational, cross-multiplication and reduction](/en/derivations/derivation-mko-rational-number-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Using the MKO's definition (there are integers p, q with q≠0 and x=p/q), prove that 6/(−8), −3/4 and 9/(−12) represent the same rational number, and reduce 6/(−8) exactly to the canonical form −3/4; prove 2/3≠3/5 by cross-multiplication; show that '5/0' denotes no number; show that the finite decimal −1.25=−5/4 and the integer 7=7/1 are rational; and, with positive denominators, prove −3/4<−2/3 by cross-multiplication. Rational number derivation-mko-rational-number-01 Concept guide ## [Real number: proving in the Cauchy completion that 'repeating nines' and the rational 1 are the same real](/en/derivations/derivation-mko-real-number-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the reals constructed as equivalence classes of Cauchy sequences of rationals (a rational q embedded as the class ι(q) of its constant sequence), let a_n=1−10^{−n} (n≥1), the finite decimals 0.9, 0.99, 0.999 and so on. Prove that (a_n) is a Cauchy sequence equivalent to the constant sequence 1, so [(a_n)]=ι(1); prove that ι is injective, so the real denoted by the finite decimal 0.999=999/1000 is not 1; and show that the finite decimal 0.375=3/8 is rational and is ι(3/8) in ℝ. Real number derivation-mko-real-number-01 Concept guide ## [Arithmetic operations: whether the same inputs give a defined result in ℕ, ℤ and ℚ](/en/derivations/derivation-mko-arithmetic-operations-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Take a=3 and b=5. In each of the number systems ℕ (including 0), ℤ and ℚ, decide whether a+b, a·b, a−b and a÷b are defined; when they are, compute the result and decide which systems it belongs to. Then decide whether 3÷0 and 0÷0 are defined in these three systems. Arithmetic operations derivation-mko-arithmetic-operations-01 Concept guide ## [Variables and expressions: a value depends on the assignment, the scope and the number system](/en/derivations/derivation-mko-variable-expression-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Let E=x+1, F=(x+1)÷2 and H=Σ_{x=1}^{3}(x+1). (1) In ℤ, evaluate E at x=2 and at x=5 and prove that E has no fixed value independent of x; (2) evaluate F at x=4 in ℚ and in ℤ; (3) evaluate H when x=5 is assigned from outside; (4) decide whether (x+1)+x and 2x+1, and whether x·x and 2x, are equivalent on ℤ; (5) distinguish the expression x+1 from the equation x+1=3 and find the solution set of the equation in ℤ. Variables and algebraic expressions derivation-mko-variable-expression-01 Concept guide ## [Equation: solving x²/(x−1)=1/(x−1) inside its domain](/en/derivations/derivation-mko-equation-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) With allowed domain ℝ, find the solution set of the equation x²/(x−1)=1/(x−1), and explain why the equation x²=1 obtained by multiplying both sides by x−1 has two solutions while the original has only one. Equation derivation-mko-equation-01 Concept guide ## [Sets: characterized by elements, equal by extensionality, and why unrestricted formation fails](/en/derivations/derivation-mko-set-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the universe ℤ, let A={x∈ℤ ∣ x²=4}, B={−2,2}, C={2,−2,2} and D={2}. Prove A=B and C=B; prove D≠B and distinguish the element 2 from the set {2}. Finally prove that in any domain U with a membership relation there is no R∈U with 'y∈R ⇔ y∉y' for all y∈U, so not every condition forms a set. Set derivation-mko-set-01 Concept guide ## [Proposition: deciding which candidate sentences are propositions, and separating 'is a proposition' from 'is proved true'](/en/derivations/derivation-mko-proposition-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In an introductory classical context, decide for each candidate whether it is a proposition, and for each proposition decide whether it is true, with reasons: s1 '7 is prime'; s2 '4 is prime'; s3 'Is 7 prime?'; s4 'Compute 3+4.'; s5 'x>0' (x an unassigned integer variable), together with closed versions of s5: substituting x=2 and x=−1, prefixing ∀x∈ℤ, and prefixing ∃x∈ℤ. Proposition derivation-mko-proposition-01 Concept guide ## [Logical connectives: testing ∨, →, ↔ and brackets on divisibility statements](/en/derivations/derivation-mko-logical-connectives-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) For an integer n let P(n) be '4∣n' and Q(n) be '2∣n'. (1) Decide the truth of '4∣12 ∨ 6∣12' and explain that disjunction is inclusive; (2) prove ∀n∈ℤ (P(n)→Q(n)); (3) prove that ∀n∈ℤ (Q(n)→P(n)) is false, and hence that ∀n∈ℤ (P(n)↔Q(n)) is false; (4) decide the truth of P(3)→Q(3) and of '2+2=4 → 7 is odd', explaining that implication is not causation; (5) compare the truth of ¬Q(2)∧P(2) and ¬(Q(2)∧P(2)). Logical connectives derivation-mko-logical-connectives-01 Concept guide ## [Mathematical proof: proving n²+n is always even, and refuting 'n²+n+41 is always prime' by a counterexample](/en/derivations/derivation-mko-proof-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) (1) In a stated context for the integers, prove from explicit premises and rules the theorem T: for every integer n, 2∣(n²+n); then derive from T by contradiction that no integer n makes n²+n odd. (2) Test the claim C: 'for every natural number n, n²+n+41 is prime'; explain why the 40 positive cases n=0 to 39 do not constitute a proof, and refute C by the counterexample n=40. Mathematical proof derivation-mko-proof-01 Concept guide ## [Set membership: x∈A is a proposition, kept apart from ⊆, = and a program's membership test](/en/derivations/derivation-mko-set-membership-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In a typed context let A={1,2,3}, B={1,2}, 𝓕={{1,2},{3}} (a set whose elements are sets of integers) and E={n∈ℤ ∣ 2∣n}. Decide with reasons whether each holds: 2∈A; 4∈A; B⊆A; A⊆B; B∈𝓕; {1}∈𝓕; 1000000∈E; 7∈E. Also explain why 1∈B and B∈𝓕 do not give 1∈𝓕, and why a programming language's membership test cannot replace the semantics of set membership. Set membership derivation-mko-set-membership-01 Concept guide ## [Subset: proving inclusion from an arbitrary element and refuting it by a counterexample](/en/derivations/derivation-mko-subset-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In the universe ℤ, let A be the set of multiples of 6, B the set of even integers and C the set of multiples of 4. Prove A⊆B and that this inclusion is strict (A≠B); prove A⊄C even though A and C share the element 12; note that B⊆B holds (⊆ allows equality); and prove {6}⊆A ⇔ 6∈A while separating the type roles of the two statements. Subset derivation-mko-subset-01 Concept guide ## [Set operations: union, intersection, difference and complement in an explicit universe](/en/derivations/derivation-mko-set-operations-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Universe U={1,2,3,4,5,6,7,8}, A={2,4,6,8}, B={3,4,5,6}. Compute A∪B, A∩B, A∖B, B∖A, U∖A and U∖B element by element; show A∖B≠B∖A; show that the complement of A changes when the universe is U′={1,2,3,4,5,6,7,8,9,10}; prove U∖(A∪B)=(U∖A)∩(U∖B) for arbitrary sets and check it on this example; and use A∪B=A∪(B∖A) to explain why the union has 6 elements, not 8. Set operations derivation-mko-set-operations-01 Concept guide ## [Sample space: two observation scales for two coin tosses, and outcomes that are not equally likely](/en/derivations/derivation-mko-sample-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Random experiment: toss a coin twice. (1) Recording the results in order, write the sample space S, and prove that it contains every possible elementary outcome and has exactly 4 elements; (2) recording only the number of heads, write the sample space S′={0,1,2}; (3) under a separately given probability model P (each ordered outcome in S has probability 1/4), compute the probabilities 1/4, 1/2, 1/4 of the outcomes in S′, showing that outcomes in one sample space need not be equally likely; (4) use another valid, non-uniform model Q on S to show that the sample space by itself does not fix probabilities; (5) distinguish the outcome (H,T) from the event {(H,T),(T,H)}. Sample space derivation-mko-sample-space-01 Concept guide ## [Event: deciding what counts as an event under the full event family and under a coarser one](/en/derivations/derivation-mko-event-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Roll a die once, S={1,2,3,4,5,6}. (1) In the finite elementary model F=𝒫(S), show that E={2,4,6} is an event, and decide whether E occurs when the observed outcome is 4 and when it is 3; distinguish the outcome 4 from the event {4}; describe the empty and the sure event. (2) In the general model (S,F₀) with F₀={∅,O,E,S} and O={1,3,5}: prove that F₀ is an event family (σ-algebra) on S, and prove {6}∉F₀, so although S is finite and {6}⊆S, {6} is not an event in this model. Event derivation-mko-event-01 Concept guide ## [Function mapping: deciding whether a rule is a function, and separating codomain, range and graph](/en/derivations/derivation-mko-function-mapping-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) Let X={−2,−1,0,1,2}, Y={0,1,2,3,4,5} and f:X→Y, f(x)=x². Prove that f is a function; compute the range f(X)={0,1,4} and show f(X)≠Y; write down the graph Γ_f; and show that changing the codomain to {0,1,4} gives a different function. Then decide two non-examples: the reverse rule s(y)='the x in X with x²=y' is not a function Y→X, and g(x)=x+4 is not a function X→Y. Function mapping derivation-mko-function-mapping-01 Concept guide ## [Tends-to relation: x→a is not x=a, and does not guarantee that function values converge](/en/derivations/derivation-mko-tends-to-relation-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D1) In (ℝ, d(x,y)=|x−y|): (1) prove that the sequence x_n=1+1/n tends to 1 while every term satisfies x_n≠1; (2) for g(x)=1 (x>1) and g(x)=0 (x≤1), prove that g(x_n)→1 along x_n, but lim_{x→1} g(x) does not exist. This shows that x→1 by itself neither requires x to equal 1 nor decides whether function values converge. Tends-to relation derivation-mko-tends-to-relation-01 1 [2](/en/derivations/?page=2#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=2#results) 299 results · Page 1 of 13