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. 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. A AB Example: A={1,2,3}, B={2,3,4} AUB = {1,2,3,4} U

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

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

7. 1.7 Set Operations A U 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}

8. A AB 1.7 Set Operations Set identities: U 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 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} Example: Ai = {i,i+1,i+2....} Examples on white board

11. 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. 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 co-domain pre- image A B f a b= f(a) domain 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) AS B f(S) f

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. A B f it is not allowed that two arrows point to the same element in B Example: f:Z  Z, f(x)=x^2 one-to-one? No; x=-1 & x=1 map both to f(1)=f(-1)=1.

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

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. A B f y x There is no element without incoming arrows Example: F:Z  Z, f(x)=x^2 onto? No, y=-1 has no pre-image.

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. A B f 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!

17. 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. A B inverse of f A B f If a function is not a bijection it is not invertible: example: f:r  R, f(x)=x^2.

18. 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 B C g C f AB Examples on white board