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.