Chapter 3 Sets and Relations
For integers, relations _____ and _____ define partial orders.
< and <=
Direct proof
Argument by deduction is just a 'logical explanation'
Power set
Denoted with 2^S is the set of all possible subsets for S Example S = {a,b,c} Power set is: {NULL,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}
Bag
Denoted with [ ] Is a collection of elements with no order (like a set), BUT with duplicate-valued elements. Example bag[3,4,5,4]
Recurrence Relation
Equation that defines a sequence based on a rule that gives the next term as a function of the previous term(s).
Written n! n being an integer greater than 0. Example 5! = 1*2*3*4*5 = 120 this is a ...
Factorial function
Proof by Contradiction
First assume that the theorem is false and then show it is true
Equivalence Relation
If a set is reflexive, symmetric, and transitive
For natural numbers what is the properties of <?
Irreflexive (cause aRa is never true), antisymmetric and transitive
Induction hypothesis
Is the case n = k of the statement we seek to prove ("P(k)"), and it is what you assume at the start of the induction step.
Mathematical Induction
Mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3).
Permutations is...
Members of S arranged in some order
A binary relation is called a _________ if it is antisymmetric
Partial order
Summations
Simply the sum of costs for some function applied to a range of parameter values.
Operation < and <= is a __________ because, for every integer pair x and y such that x does not equal y, either x < y or y < x
Total order
Boolean variables
Variable that takes on values True = 1 and False = 0
R is Reflexive if...
aRa for all a E S Simple Terms: For every real number x, x=x
R is Irreflexive if...
aRa is not true for all a E S Example: A = {a,b} then R = {(a,b),(b,a)} is a irreflexive relation
R is Symmetric if...
whenever aRb and bRa, then a = bfor all a, b E S Simple Terms: For all real numbers x and y if x = y, then y = x
R is antisymmetric if...
whenever aRb and bRa, then a=b, for all a, bES Example : A = {1,2,3,4} Antisymmetric will be R= {(1,1),(2,2),(3,3),(4,4)}
R is Transitive if...
whenever aRb and bRc, then aRc for all a,b,c E S Simple Terms: if x = y and y = z, then x = z
Logarithms
Base b for value y is the power to which b is raised to get y Written log(little b ) y = x then b^x = y and b^log(little b)y = y
Members drawn from some larger population known as ______________
Base type (example set of integers, natural numbers etc.)
Set
Collection of distinguishable members of elements Example: {2,4,5,7}
Sequence/tuple/vector
Collection of elements with order denoted with <> Example <3,4,5,4>
Elements x and y of a set are ________ under a given relation if either xRy or yRx
Comparable
For powerset of integers, the subset operator defines a _____________ (antisymmetric and transitive)
Partial order
The set on which partial order is defined is called a ___________
Poset aka Partially ordered set
Proving Contrapositive
Prove P -> Q by proving ~Q -> ~P
Induction step
Proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1.
Base Case
Proves the statement for n = 0 without assuming any knowledge of other cases
Relation
R over set S is a set of ordered pairs from S. Examples: S = {a,b,c} then a relation is {<a,c>,<b,c>,<c,d>}
Random Permutation
Random ordering of a set of objects.
For natural numbers what is the properties of <=?
Reflexive, antisymmetric and transitive
For natural numbers what is the properties of =?
Reflexive, symmetric (antisymmetric) and transitive.
Closed-form solution
Replacing summation with algebraic equation with same value
If relation is irreflexive it is called a ________
Strict partial order
Ceiling [` `]
Takes and resturns greatest integer Example [` 3,4 ] = 4
Floor [_ _]
Takes x and returns the least integer Example: [_ 3,4 _] = 3
What is the meaning of xRy?
This is notation to show that tuple <x,y> is in relation
If every pair of distinct elements in a partial order are comparable, then the order is called a _________ or __________
Total order Linear order
Estimation
When we use approximate values in a calculation to give an approximate.
