Essential Mathematical Concepts: Sets, Relations, and Estimation Techniques
This chapter provides fundamental mathematical definitions and concepts, including sets, bags, sequences, and relations. It explains the properties of relations such as reflexivity, symmetry, and transitivity, and introduces equivalence relations and partial orders. The text also covers important notions of notation for computer science, like bits and bytes, and introduces concepts of recursion and proof strategies. Lastly, it illustrates estimation through a practical example of fitting golf balls in a room, demonstrating key parameters and calculations for problem-solving.
Essential Mathematical Concepts: Sets, Relations, and Estimation Techniques
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Presentation Transcript
Chapter 2 Mathematical Preliminaries
Sets and Relations • A set is a collection of distinguishable members or elements • The members are usually drawn from some larger base set • Each member is either a primitive element of the base set or a set itself • The is no concept of duplication in a set • Each value from the base type is either in the set or not • Example 3, 4, 5 are in set P and the base type is ints
Bags • Sometimes we wish to define a collection without order, like a set, but with duplicate items • Such a collection is called a bag • To distinguish a bag from a set we put square brackets around a bag’s elements
Sequences • A sequence is a collection of elements with an order and which may contain duplicate-value elements. • A sequence is also sometimes called a tuple or a vector • A sequence is indicated using angle brackets <>
A relation R over set S is a set of ordered pairs from S If tuple <x,y> is in relation R, we can show it as xRy We can define the following properties of relations: R is reflective if aRa for all a in S R is symmetric if whenever aRb, then bRa for all a,b in S R is antisymmetric if whenever aRb and bRa, then a=b for all a,b in S R is transitive if whenever aRb and bRc then aRc for all a,b,c in S Relation
Equivalence Relation • R is an equivalence relation on set S if it is reflexive, symmetric and transitive • An equivalence relation can be used to partition a set into equivalence classes • An equivalence relation on set S partitions the sets into subsets whose elements are equivalent
Partial Order • A binary relation is called a partial order if it is antisymmetric and transitive. • The set on which the partial order is defined is called a partially ordered set or a poset
Miscellaneous Notation • B means bytes • b means bits • KB is a kilobyte 210 = 1024 byes • MB is a megabyte 220 bytes • GB is a gigabyte 230 bytes
n! • The factorial function for integer n is the product of the numbers between 1 and n • Stirling’s approximation is n! ≈ √2πn(n/e)n
Recursion • Recursion is awesome!! • Two parts • Base case • Recursive call • You’ll see a lot more of recursion this semester
Proof Strategies • Proof by contradiction • Proof by Mathematical Induction • Base case • Inductive Hypothesis • (seem familiar)
Estimating • Estimating can be formalized by a three step process • Determine the major parameters that affect the problem • Derive an equation that relates the parameters to the equation • Select values for the parameters, and apply the equation to yield an estimated solution
Example • How many golf balls can fit in room that is 8” x 8” x 8” • Parameters: a golf ball is 1’ diameter • The room is 4096 cubic feet • You can fit 1728 golf balls in 1 cubic foot • So you can fit 4096 * 1728 total golf balls • 7,077,888 golf balls • The units do to match!!
