1 / 20

Lecture 3

Lecture 3. Set Operations & Set Functions. Recap. Set: unordered collection of objects Equal sets have the same elements Subset: elements in A are also in B Power-set: set of all subsets Cartesian Product: set of ordered pairs. A. A. B. 1.7 Set Operations.

tomai
Télécharger la présentation

Lecture 3

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. Lecture 3 Set Operations & Set Functions

  2. Recap • Set: unordered collection of objects • Equal sets have the same elements • Subset: elements in A are also in B • Power-set: set of all subsets • Cartesian Product: set of ordered pairs.

  3. A A B 1.7 Set Operations Union:The union of 2 sets A and B is the set with elements in either A or B or both. U Example: A={1,2,3}, B={2,3,4} AUB = {1,2,3,4}

  4. A A B 1.7 Set Operations Intersection: The intersection of 2 sets A and B is the set containing elements in both A and B. Example: U Example: A={1,2,3}, B={2,3,4} A B = {2,3}

  5. 1.7 Set Operations Sets are disjoint if their intersection is the empty set. I.e. They have no elements in common. Principle of inclusion-exclusion: this terms is needed because elements both in A and B are counted once in AUB. Example: A={1 2 3}, B = {p,q,r}. A and B are disjoint. |AUB|=|A|+|B|=6 Example: A={1 2 3}, B={1 2 3}. A=B. |AUB|=3+3-3=3.

  6. A A B 1.7 Set Operations Difference: The difference between 2 sets A and B is the set with elements in A but not in B. U Example: A={1 2 3}, B={2 3 4} A-B={1}

  7. 1.7 Set Operations Complement: The complement of a set A in U is the set U-A. Example: U={x|x in English alphabet} A={x| x is consonant } A = {x|x is vowel} U A

  8. 1.7 Set Operations Set identities: U A A B Morgan’s law

  9. 1.7 Set Operations Membership tables (just like truth tables): Generalized Union: The union of a collection of sets is the set with elements in at least one of the sets in the collection. Generalized intersection:The intersection of a collection of sets is the set with elements in all sets in the collection.

  10. 1.7 Set Operations Example: Ai = {i,i+1,i+2....} Efficient representation in a computer: Assume arbitrary order and denote set membership with a bit string: U={1,2,...10} (in this order) A = {1 2 3 4 5}=1 1 1 1 1 0 0 0 0 0 B = {4,5,6} = 0 0 0 1 1 1 0 0 0 0 AUB = 1 1 1 1 1 1 0 0 0 0 = BIT-wise AND = {1 2 3 4 5 6}

  11. domain f a b= f(a) image A B 1.8 Functions function: The assignment of exactly one element of the set B to each element of the set A. f:AB or f(a)=b. co-domain A is the domain of f. B is the co-domain. of f. b is the image of a. a is the pre-image of b. range of f: set of all images of elements of A. range pre- image Example: f:ZZ, f(x)=x^2 domain/co-domain: Z range: perfect squares {0,1,4,9,...}

  12. 1.8 Functions If f1 and f2 are two functions from A to R (real numbers), then g=f1+f2 and h=f1*f2 are also functions defined by: (f1+f2)(x) = f1(x) + f2(x) (f1*f2)(x) = f1(x)*f2(x) Example: f1(x) = x, f2(x) = x^2. (f1+f2)(x) = x+x^2 (f1*f2)(x) = x^3. f:AB, and S is a subset of A. Then we can define f:SImage(S)=f(S) f f(S) A S B

  13. 1.8 Functions One-to-one or injective function: A function f is one-to-one if and only if f(x)=f(y) implies x=y for all x,y in domain f. f it is not allowed that two arrows point to the same element in B equivalent since: Example: f:ZZ, f(x)=x^2 one-to-one? No; x=-1 & x=1 map both to f(1)=f(-1)=1. A B

  14. 1.8 Functions strictly increasing strictly decreasing x,y real decreasing increasing f(x) > f(y’) f(y) > f(a)=f(b) f(x’) y x < x’ < y’ Strictly increasing and strictly decreasing functions are one-to-one

  15. 1.8 Functions Onto or surjective functions: A function f from A to B is onto if for every element b in B there is an element a in A with f(a)=b. x y f Example: F:ZZ, f(x)=x^2 onto? No, y=-1 has no pre-image. A B There is no element without incoming arrows

  16. 1.8 Functions One-to-one correspondence or bijection: A function f is in one-to-one correspondence if it is both one-to-one and onto. Number of elements in A and B must be the same. Every element in A is uniquely associated with exactly one element in B. Example: f:RR, f(x)= -x bijection! f A B

  17. f A B 1.8 Functions Inverse function: The inverse function of a bijection is the function that assigns to b in B the element a in A such that f(a)=b. inverse of f A B If a function is not a bijection it is not invertible: example: f:rR, f(x)=x^2.

  18. g f C 1.8 Functions Composition: A composition of 2 functions g:AB and f:BC is defined by: Range of g must be subset of domain of f! A C B A B

  19. 1.8 Functions Graph of a function: The graph of a function f is the set of ordered pairs {(a,b)|a in A, f(a)=b}. This is a subset of the Cartesian product AXB (i.e. it is a “relation”). f(a) a in A (for some ordering)

  20. 1.8 Functions Some examples: (more details later)

More Related