Sets Defined

1. Sets Defined • A set is an object defined as a collection of other distinct objects, known as elements of the set • The elements of a set can be anything: people, plants, numbers, functions, and even other sets. • Using sets, nearly any mathematical concept can be derived

2. Describing Sets • There are many accepted ways to describe sets. • Intensional Definition: A is the set of all integers • Extension: using braces – C = {red, blue, green} • Use “…” as a shorthand – D = {1,2,3,…,1000} • Set-Builder notation: using a formula • F = {n^2: n is an integer; and 0 ≤ n ≤ 19} • This is the set of the first 20 squared integers • The “:” means “Such That” • This form can be expanded by indicating the set of all elements with a given property

3. Membership • The key relation between sets is membership • If set X is a member of set Y, then all of the elements of X are also in Y • This is loosely denoted X Y, where X is a subset of Y • If element a is a member of set Y, we say aY

4. Subsets • Do not be mislead, there is a proper way to denote subsets • If X is a subset of Y, we say . The second notation is read Y is the superset of X, meaning that Y contains X • If we know that X is a subset of Y, but X does not equal Y, then X is a proper subset of Y, denoted

5. Set Equality • A useful way to define the equality of sets is to say , meaning X and Y are subsets of each other (their elements are equal) • Because a set is a collection of distinct objects, the order of elements and the frequency of the same element are meaningless • {11,7} = {7,11} = {11,7,11,11,7}

6. Special Kinds of Sets • A null set (Ф)is defined as a set that is negligible in some sense • For example, if an object’s length is to be measured, the value 0 would be negligible, meaning that if an object has a length of 0, it has no length at all • An empty set is always a null set, in every situation • In general, if the value of the elements of a set can be disregarded, the set is a null set • Consider a set used to represent integers. If only integers matter, then {a,b,c} is a null set • A universal set (U) is the set that contains all meaningful objects, including itself • This set is very controversial, and in many cases is said not to exist • It is basically the opposite of the null set

7. Basic Set Operations • The two most common set operations are Union and Intersection • The Union of a collection of sets is a derived set containing all of the elements in either of the sets • The Intersection of a collection of sets is a derived set containing all of the elements in each of the sets • For example, the union of the sets {1,2,3} and {3,4,5} is the set {1,2,3,4,5} and the intersection is the set {3}

8. Properties of Sets • The following are useful properties of sets and set operations • Inclusion of Intersection: • Inclusion of Union: • Transitivity: • Commutative Laws: • Associative Laws: • Distributive Laws:

9. Set Partitions • Partitions of a set are subsets which do not share any elements • These sets are said to be disjoint, meaning the intersection of any two partitions is the null set • Partitions are very useful in computer science. A hard drive can partitioned into multiple drives, enabling certain information to be stored on one partition and other information on another partition

10. Difference and Complement • The difference of two sets (B-A) is the set of all elements x in U such that x is in B but not in A • The complement of a set is the set of all elements that are not in the original set

11. More Set Properties • If an element is in a set’s complement, it is not in the original set • Identity Laws: • Complement Laws: • Double Complement Law: • Idempotent Laws: • Universal Bound Laws: • DeMorgan’s Laws: • Absorption Laws: • Special Set Complements: • Set Difference Law:

12. Visual Aids • Set operations can be represented by visual aids like Venn Diagrams • Given sets A={1,2} and B={2,3} and outside value (4) • “A union B” is • “A intersect B” is • “A complement” is • “A – B” is • “Complement (A union B)” is • “Complement (A intersect B)” is • Any set operation can be represented by a visual aid, regardless of complexity. This is useful in proofs!

13. Power Sets • The power set of set A is the set of all subsets of A • Consider the set {1,2} • The power set of {1,2} contains all subsets, so the null set and the original set itself are also elements • This is denoted • It has been proven (through induction) that a set of n elements has exactly subsets

14. Tuples • A set, such as {1,2} is equal to all other sets that contain only the elements {1,2} in any order and frequency, such as {1,2,1,1} • The opposite is true of an ordered tuple • The ordered tuple (x[1], x[2],…,x[n]) is equal to the ordered tuple (y[1], y[2],…,y[n]) iff “x[1] = y[1], x[2] = y[2], …,x[n] = y[n]” • Each element must equal its partner – (a,b) = (c,d) iff “a = c and b = d” • An ordered 2-tuple is denoted ((x[1],y[1]),…,(x[n],y[n])) and is called an ordered pair. A 3-tuple is an ordered triple

15. Cross Products (a.k.a. Cartesian Products) • The cross product of A and B (AxB) is the set of all ordered pairs (a,b) where a is in A and b is in B • The cross product of {1,2,3} and {a,b} is {(1,a), (2,a), (3,a), (1,b), (2,b), (3,b)} • The cross product of n number of sets is a set of ordered tuples, with each tuple containing n elements

16. Procedural Definitions • In order to use these concepts in CS, you must be able to use their definitions to generate procedures • Union: if element x is in “A union B”, it must be in A or B • If x is in “A intersect B”, it must be in A and B • If x is in “A difference B”, it must be in A and can’t be in B • If x is in “A complement”, it cannot be in A • If (x,y) is in “A cross B”, x must be in A and y must be in B • If A is a subset of B, every element a that is in A must be in B • If set A and B are equal, A must be a subset of B and B must be a subset of A • From these procedures, set identities and laws can be constructed in your code. Try extending set theory definitions to more than two sets. Happy coding!

17. Subset Algorithm • This common algorithm is used to prove subset relations • Use this algorithm to help define other algorithms. Remember: reusing code is a good thing