1 / 12

Discrete Mathematical

Explore discrete mathematical examples, nested quantifiers, and the meaning of multiple quantifiers using predicate logic to understand order and implications. Learn how to determine the validity of arguments.

jeffryf
Download Presentation

Discrete Mathematical

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Discrete Mathematical

  2. Example OR Q(x,y): x+y=x-y a) Q(1,1): 2=0 False b) Q(2,0): 2+0=2-0 True c) Q(1,y): 1+y=1-y False(take any y<>0, x: y=1) d) Q(x,2): x+2=x-2 False

  3. First: SOLUTION Q1. (5 pts) Show that the following argument form is invalid:

  4. Q2. Use the truth table to show if the argument is valid. " If this number is larger than 2, then its square is larger than 4." " This number is not larger than 2. " The square of this number is not larger than 4. p → q  p  q

  5. True proposition • False proposition • Not a proposition • No clue c) b) a) b) Predicates - multiple quantifiers (Nested quantifiers) To bind many variables, use many quantifiers! Example: P(x,y) = “x > y” • x P(x,y) • xy P(x,y) • xy P(x,y) • x P(x,3)

  6. P(x,y) true for all x, y pairs. For every value of x we can find a (possibly different) y so that P(x,y) is true. P(x,y) true for at least one x, y pair. There is at least one x for which P(x,y) is always true. quantification order is not commutative. Predicates - the meaning of multiple quantifiers • xy P(x,y) • xy P(x,y) • xy P(x,y) • xy P(x,y)

  7. False True? True False Predicates - the meaning of multiple quantifiers N(x,y) = “x is sitting by y” • xy N(x,y) • xy N(x,y) • xy N(x,y) • xy N(x,y)

  8. Multiple quantifiers (Examples) • " x " y, P(x,y): • For all x and for all y the relation P(x,y) is true. • If two numbers are integers then their product is an integer. • 2. " x $ y, P(x,y): • For all x there is some y such that P(x,y) is true. • Every student has a favorite teacher • Note: here and below in all examples concerning people, we shall assume that the domain is known and will not represent it neither separately, nor within the predicate expression.

  9. Multiple quantifiers (Examples) 3. $ x " y, P(x,y): There is some x such that for all individuals y the relation P(x,y) is true. Someone is loved by everybody $ x " y loves (y,x) There is a professor that is liked by all students 4. $ x $ y, P(x,y): There is some x and there is some y such that P(x,y) is true. Some students have favorite teachers

  10. Extra exmples for multiple quantifiers • xy P(x, y) • “For all x, there exists a y such that P(x,y)” • Example: xy (x+y == 0) • xy P(x,y) • There exists an x such that for all y P(x,y) is true” • Example: xy (x*y == 0)

  11. Order of quantifiers • xy and xy are not equivalent! • xy P(x,y) • P(x,y) = (x+y == 0) is false • xy P(x,y) • P(x,y) = (x+y == 0) is true

More Related