# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Sets: characterized by elements, equal by extensionality, and why unrestricted formation fails](/en/derivations/derivation-mko-set-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-set)

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



## [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%3Fq%3Dmko-set)

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



## [Set operations: union, intersection, difference and complement in an explicit universe](/en/derivations/derivation-mko-set-operations-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-set)

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





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