Review of Mathematical
Notation / Terminology
Sets, Venn Diagrams, Sequences,
Tuples, Functions, Relations, Graphs,
Strings, Languages, Boolean Logic
Sets
• Order doesn’t matter
– {7, 6, 5} and {5, 6, 7} are the same.
• In a set, repeats are “not allowed”
– {7, 7} is really {7}, i.e., they describe the same set.
• In a multiset, repeats are allowed
– {7, 7} and {7} are different
Sets
•
•
•
•
•
Empty set notation?
Union
Intersection
Compliment
Set Difference?
Venn Diagrams
• Starts with…
• Ends with..
• Contains…
• Questions…
Sequences
• Like sets, but the order matters and repeats
are “allowed”
• (5, 4, 7) is a different sequence than (4, 5, 7),
but they would be the same set.
• (5, 5, 5, 6) is a different sequence than (5, 5, 6)
but they are the same set.
Tuples
• Its just another way of describing sequences.
– 2-tuple is a pair
– 3-tuple is a trio
• Question: If A = {1,2} and B= {x,y,z} what is
A X B?
– X is the Cartesian product.
– Note: This will create a set of pairs, 2-tuples, or
sequences of size 2.
Power Set
• A = {0, 1, 2}
• Power set of A is
• { {}, {0}, {1}, {2}, {0,1}, {1,2}, {2,0}, {0,1,2}}
• “Power Sequence” of A is
• { (), (0), (1), (2), (0,1), (1,0), (1,2), (2,1)…
(0,1,2), (1,2,0), (2,0,1), (2,1,0), …)
• Question: What is the size of the set above?
Functions
• f(a) = b
• Also called a mapping
• Function: Domain  Range
– Abs: Z  Z
– Add: Z X Z  Z
– Division: Z X Z  Rational Numbers
• Question: Example 0.8, 0.9, and 0.10
Relation
• Function whose Range is {TRUE, FALSE} is called a
Predicate
• Predicate whose Domain is a tuple is called a
Relation.
• If the Domain is a 2-tuple or pair, then its called a
Binary Relation
• Example: Equality of two numbers
–
–
–
–
Java: a == b or a.equals(b)
f(a,b) = true if a equals b, otherwise false
aRb, where R is the equality Relation
F: Z X Z  {TRUE, FALSE}
Equivalence Relation
• Satisfies three conditions
1. Reflexive: xRx is always true
2. Symmetric: if xRy is true, then yRx is true
3. Transitive: if xRy and yRz are true, then xRz is true.
• Problems: Are the following Relations
equivalence relations:
– Equality x == y
– Less-than x < y
– F(x,y) = true if x+y is even, otherwise false
Graphs
•
•
•
•
•
•
•
Directed vs. undirected
Nodes/vertices
Edges
Degree
Labeled graph
Sub-graph
Path
•
•
•
•
•
•
Cycle
Simple cycle
Tree
Root node
Leaf nodes
Strongly connected
directed graphs
Languages
•
•
•
•
•
•
Alphabet notation
No quotes
Empty string
Substring
Concatenation
Lexiographic ordering
Boolean Logic
•
•
•
•
•
And
Or
Not
XOR
Distributive law