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EECS 20 Lecture 2 (January 19, 2001) Tom Henzinger

Mathematical Language. EECS 20 Lecture 2 (January 19, 2001) Tom Henzinger. Mathematical Language. Let Evens = { x | (  y, y  Nats  x = 2· y ) }. Mathematical Language. Let Evens = { x | (  y, y  Nats  x = 2 · y ) }. Constants. Mathematical Language.

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EECS 20 Lecture 2 (January 19, 2001) Tom Henzinger

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  1. Mathematical Language EECS 20 Lecture 2 (January 19, 2001) Tom Henzinger

  2. Mathematical Language Let Evens = { x | (  y, y  Nats  x = 2· y ) } .

  3. Mathematical Language Let Evens = { x | (  y, y  Nats x = 2· y ) } . Constants

  4. Mathematical Language Let Evens = { x | (  y, y Nats x = 2· y ) } . Constants Variables

  5. Mathematical Language Let Evens = { x | (  y, y Nats  x = 2· y ) } . Constants Variables Operators

  6. Mathematical Language Let Evens = { x |(  y, y  Nats  x = 2· y )} . Constants Variables Operators Quantifiers

  7. Mathematical Language Let Evens = { x | (  y, y  Nats  x = 2· y ) } . Constants Variables Operators Quantifiers Definition

  8. Mathematical Language LetEvens= { x | (  y, yNatsx=2· y ) } . Constants Variables Operators Quantifiers Definition

  9. Constants have meaning 20 a certain number Berkeley a certain city false a certain truth value

  10. Variables have no meaning x y0 z’

  11. Operators on numbers number + number Result: number number ! number number = number truth value numbernumber truth value

  12. Operators on cities merge ( city, city ) Result: city population-of ( city ) number has-a-university ( city ) truth value

  13. Operators on truth values truth value  truth value Result: truth value truth value  truth value truth value ¬ truth value truth value truth value truth value truth value truth value truth value truth value

  14. Expressions on constants have meaning 3 + 20 Result: 23 (3! + 2) · 4 32 4  population-of ( Berkeley ) true 4 · 20  4 + 20 false true  false false true  ( 4 + 20 ) not well-formed

  15. Implication true  true Result: true true  false false false  true true false  false true

  16. Expressions on variables have no meaning x + 20 (3! + y) · 4 x  y Free variables:x y x, y

  17. Quantifiers remove free variables from expressions x = 0  x, x = 0  x, x = 0  y, x + 1 = y  x,  y, x + 1 = y  x,  y, x  y  x, x + 7 Result: free x true false free x true true not well-formed

  18. Every mathematical expression 1. is not well-formed (“type mismatch”), or 2. contains free variables, or 3. is a definition, or 4. has a meaning (e.g., 20, Berkeley, false).

  19. SETS

  20. Set constants { 1, 2, 3 } { Atlanta, Berkeley, Chicago, Detroit } { 1, 2, 3, 4, … }

  21. Set operator anything  set Result: truth value 2  { 1, 2, 3 } true 2  { Atlanta, Berkeley } false

  22. Set quantifier (  x, truth value ) Result: truth value (  x, truth value ) truth value { x | truth value } set

  23. Quantifiers remove free variables from expressions { x | x  y } { x | x = 1  x = 2 } { x |  y, x = 2· y } { x | x + 7 } Result: free y { 1, 2 } { 2, 4, 6, 8, … } not well-formed

  24. Bounded quantification (  x set, truth value ) Result: truth value (  x set, truth value ) truth value { x set | truth value } set

  25. Meaning of constants can be defined Let Nats = { 1, 2, 3, 4, … } . Let Bools = { true, false } . Define Cities = { Atlanta, Chicago, Berkeley, Detroit } . Define Ø = { } .

  26. Let Evens = { x  Nats |  y  Nats, x = 2· y } . Let Evens be the set of all x  Nats such that x = 2· y for some y  Nats.

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