CHAPTER III/IV Discrete Mathematics

Subset

A set Y is a subset of set X (Written as X ⊆ Y) if every element of X is an element of set Y. Example − Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set X is a subset of set Y as all the elements of set X is in set Y. Hence, we can write X ⊆ Y.

Finite Set

A set which contains a definite number of elements. Example − S = {x | x ∈ N and 70 > x > 50}

Infinite Set

A set which contains infinite number of elements. Example − S = {x | x ∈ N and x > 10}

Empty Set or Null Set

An empty set contains no elements. It is denoted by ∅. As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero. Example − ∅ = {x | x ∈ N and 7 < x < 8}

capital letters A, B, C, etc.

Generally, sets are named with the ______

Equivalent Set

If the cardinalities of two sets are same. Example − If A = {1, 2, 6} and B = {16, 17, 22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A| = |B| = 3

Path Cost

If the given is a weighted graph, it is the totality of the line segment value from the initial state to final state.

Disjoint Set

If two sets C and D are disjoint sets as they do not have even one element in common. Therefore, n(A ∪ B) = n(A) + n(B) Example − Let, A = {1, 2, 6} and B = {7, 9, 14}, there is no common element, hence these sets are overlapping sets.

Equal Set

If two sets contain the same elements they are said to be equal. Example − If A = {1, 2, 6} and B = {6, 1, 2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

Universal Set

It is a collection of all elements in a particular context or application. Example − We may define U as the set of all animals on earth. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.

Goal Test

It is a valid way from starting point up to the final state

Descriptive Form

One way to specify a set is to give a verbal description of its elements

P(S)

Power set is denoted as

Power set of a set

S is the set of all subsets of S including the empty set.

Set Union Set Intersection Set Difference Complement of Set Cartesian Product.

Set Operations include

Finite Set Infinite Set Subset Proper Subset Universal Set Empty Set or Null Set Singleton Set or Unit Set Equal Set Equivalent Set Overlapping Set Disjoint Set

Sets can be classified into many types

Cartesian Product / Cross Product

The Cartesian product of n number of sets A1, A2 .....An, defined as A1 × A2 ×..... × An, are the ordered pair (x1, x2,....xn) where x1 ∈ A1, x2 ∈ A2, ...... xn ∈ An Example − If we take two sets A = {a, b} and B = {1, 2}, The Cartesian product of A and B is written as − A × B = {(a, 1), (a, 2), (b, 1), (b, 2)} The Cartesian product of B and A is written as − B × A = {(1, a), (1, b), (2, a), (2, b)}

Ellipsis

are used to indicate that the pattern of the listed elements continues, as in { 5, 6, 7,...... } or { 3, 6, 9, 12, 15,........60 }.

Ellipsis

can be used only if enough information has been given so that one can figure out the entire pattern.

Cardinality of a set S

denoted by | S |, is the number of elements of the set. If a set has an infinite number of elements, its cardinality is ∞.

Final State

end point; symbol is the double circle of the node

set

is a collection of well defined objects.

Set-builder notation

is a notation for describing a set by indicating the properties that its members must satisfy.

Longest Path

it is the goal test w/ got most number of nodes visited (unweighted graph)

Shortest Path

it is the goal test w/ smaller number of nodes visited (unweighted graph)

List of Operators

list of nodes, successor function

Set

may be considered as a mathematical way of representing a collection or a group of objects.

Sets

often deal with a group or a collection of objects, such as a collection of books, a group of students, a list of states in a country, a collection of coins, etc.

Singleton Set or Unit Set

set contains only one element. A singleton set is denoted by {s}. Example − S = {x | x ∈ N, 7 < x < 9}

Universal

sets are represented as U.

Initial State

starting point; symbol is single circle of the node

Z

the set of all integers = {....., −3, −2, −1, 0, 1, 2, 3, .....}

N

the set of all natural numbers = {1, 2, 3, 4, .....}

Z+

the set of all positive integers

Q

the set of all rational numbers

R

the set of all real numbers

W

the set of all whole numbers

Complement of a Set

The complement of a set A (denoted by A') is the set of elements which are not in set A. Hence, A' = {x | x ∉ A}. More specifically, A'= (U − A) where U is a universal set which contains all objects. Example − If A = {x | x belongs to set of odd integers} then A' = {y | y does not belong to set of odd integers}

small letters a, b, c, etc.

The elements of a set are denoted by the ______

Set Intersection

The intersection of sets A and B (denoted by A ∩ B) is the set of elements which are in both A and B. Hence, A ∩ B = {x | x ∈ A AND x ∈ B}. Example − If A = {11, 12, 13} and B = {13, 14, 15}, then A ∩ B = {13}.

elements or members of the set

The objects of a set are called ______ or _______

distinct

The objects of a set are all

Set Difference/ Relative Complement

The set difference of sets A and B (denoted by A − B) is the set of elements which are only in A but not in B. Hence, A − B = {x | x ∈ A AND x ∉ B}. Example − If A = {10, 11, 12, 13} and B = {13, 14, 15}, then (A− B) = {10, 11, 12} and (B − A) = {14, 15}. Here, we can see (A − B) ≠ (B − A)

Set Builder Notation

The set is defined by specifying a property that elements of the set have in common.

Roster or Tabular Form

The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

Set Union

The union of sets A and B (denoted by A ∪ B) is the set of elements which are in A, in B, or in both A and B. Hence, A ∪ B = {x | x ∈ A OR x ∈ B}. Example − If A = {10, 11, 12, 13} and B = {13, 14, 15}, then A ∪ B = {10, 11, 12, 13, 14, 15}. (The common element occurs only once)

Overlapping Set

Two sets that have at least one common element. Example − Let, A = {1, 2, 6} and B = {6, 12, 42}. There is a common element '6', hence these sets are overlapping sets.