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Sets, Relations, Life, and Death

Mathematically speaking…. Sets, Relations, Life, and Death. by Pavel Gladyshev. Objects. Chair, You, Me, 1, 2, 3, UCD, pack of pringles Any two objects x and y can be compared for equality:. Set. Unoprdered c ollecton of distinct objects (members)

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Sets, Relations, Life, and Death

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  1. Mathematically speaking… Sets, Relations, Life, and Death by Pavel Gladyshev

  2. Objects • Chair, You, Me, 1, 2, 3, UCD, pack of pringles • Any two objects x and y can be compared for equality:

  3. Set • Unoprderedcollecton of distinct objects (members) • May contain other sets as members • Set names are written in capital Roman letters: A, B, N, R, …

  4. Membership

  5. Standard sets

  6. Set definition • Enumeration of elements of the set • Restriction of (i.e. selection of elements from) an existing set • Combining existing sets using set operators

  7. Set Definition by Enumeration

  8. Set Definition by Restriction • A consists of all members of B for which the logical formula is true

  9. Logical formula • Basic statements • Connected with logical connectives

  10. If no base set is specified, x is assumed to be a member of the Universal set:

  11. Subset “A is a subset of B” if every element of A is a member of B. A ` ` B

  12. Set equality

  13. Set operators • Union • Intersection • Subtraction

  14. Set operations • Union: • Intersection: • Difference (“subtraction”)

  15. Assuming that • All men are mortal • Socratis is a man • Convince me that Socratis is mortal

  16. Tuple • Ordered sequence of objects • Same object can appear in a tuple several times • Elements of a tuple are referred to with subscripts:

  17. Relation A collection of links between elements of two or more sets: Set B (Range of the retation) a 1 b 2 c 3 Set A (Domain of the retation)

  18. Relation • Formally defined as a set of tuples: • Can be viewed as predicate:

  19. Let’s talk mathematically about sexuality • Define concepts “heterosexual”, “gay”, “lesbian”, etc. • Convince me using Venn diagrams that gays are human beings.

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