Sets

o A set is an unordered collection of objects.o The objects in a set are called the elements or members of

the set S, and we say S contains its elements.o Sets can be defined by listing their elementse.g. S = {2, 3, 5, 7, 11, 13, 17, 19}, S = {CS1202, CS542, ERG2020, MAT141}

Sets

o Examples of discrete structures built with the help of sets:o Countingo Combinationso Relationso Graphs

Representing Sets by Properties

o It is inconvenient, and sometimes impossible, to define a set by listing all its elements.

o Alternatively, we can define by a set by describing the properties that its elements should satisfy.

{x| x has property P}./ {xA| P(x)}set of elements, x, in A such that x satisfies property P.{x and -2<x<5}E = {x| 50 <= x < 63, x is an even integer}

Alternative Way of Defining Sets

We can implicitly define a set using a predicate to characterize its elements.

A= {x S | P(x)},∈the set of all such thatwhich is read “the set of all x in S such that P of x.”

Define a set using predicate .For example, the set E of even numbers is

the multiples of 3 between 20 and 30

Sets

o Natural numbers:– = {0,1,2,3, …}

o Integers– = {…, -2,-1,0,1,2, …}

o Positive integers– Z+ = {1,2, 3.…}

o Rational numbers – = {p/q | p Z, q Z, q 0}

Set

o Size of a Seto The size of a set S, denoted by |S|, is defined as the number of

elements contained in S.• If S = {2, 3, 5, 7, 11, 13, 17, 19}, then |S|=8.

o NULL SET:o A set which contains no element is called a null set, or an empty set

or a void set. It is denoted by the Greek letter (phi) or { }.A = {x | x is a person taller than 10 feet} = ( Because there does not exist any human being which is taller then 10 feet )B = {x | x2 = 4, x is odd} = (Because we know that there does not exist any odd whose square is 4)

Set

o A Subset– A set A is said to be a subset of B if and only if every element of

A is also an element of B. We use A B to indicate A is a subset ⊆of B.• ∀x, if x A then x B∈ ∈

1. When A B, then B is called a superset of A.2. When A is not subset of B, then there exist at least one x A

such that x B.3. Every set is a subset of itself

is regarded as a subset of every set.

Seto Every Set has necessarily two subsets and the Set itself, these

two subset are known as Improper Subsets and any other subset is called Proper Subset.

o Given two sets A and B, we say A is a proper subset of B, denoted by , if every element of A is an element of B, But there is an element in B that is not contained in A.

Set Equalityo Given sets A and B, A equals B, written A = B, if, and only

if, every element of A is in B and every element of B is in A.

Symbolically:A = B A B and B A.⇔ ⊆ ⊆

Set Equality

o Two sets are called disjoint if their intersection is empty.o Alternate: A and B are disjoint if and only if

A B =

Defining Sets

S=T

S T

S is a proper subset of T.

Set

o UNIVERSAL SET:The set of all elements under consideration is called the Universal Set. The Universal Set is usually denoted by U.

o Finite SetIf it contains exactly m distinct elements where m denotes some non negative integer.

In such case we write |S| = m or n(S) = m

o A set is said to be infinite if it is not finite.

Set OperationsLet A and B be subsets of a universal set U.1. The union of A and B, denoted A B, is the set of all elements that are in at least one of A ∪

or B.2. The intersection of A and B, denoted A ∩ B, is the set of all elements that are common to

both A and B.3. The difference of B minus A (or relative complement of A in B), denoted B − A, is the set of

all elements that are in B and not in A.4. The complement of A, denoted Ac, is the set of all elements in U that are not in A.Symbolically:

A B = {x U | x A or x B},∪ ∈ ∈ ∈A ∩ B = {x U | x A and x B},∈ ∈ ∈

B − A = {x U | x B and xA},∈ ∈Ac = {x U | x A}.∈

Power Set

The power set of a set A is the set of all subsets of A, and is denoted by 2A. That is,2A = {S : SA}.

For example, for A = {2, 4, 17, 23}, we have2A ={, {2}, {4}, {17}, {23}, {2, 4}, {2, 17}, {2, 23}, {4, 17}, {4, 23}, {17, 23},{2, 4, 17}, {2, 4, 23}, {2, 17, 23}, {4, 17, 23}, {2, 4, 17, 23}}

The cardinality of this set is 16,

Set Identities

Functions

• A function f from a set X to a set Y,– Denoted f : X → Y , is a relation from X, the domain, to Y , the co-

domain, – that satisfies two properties:– (1) every element in X is related to some element in Y , and– (2) no element in X is related to more than one element in Y .

Functions

• This arrow diagram does define a function because

1. Every element of X has an arrow coming out of it.2. No element of X has two arrows coming out of it that point to two different elements of Y .

One-to-One Functions

• Let F be a function from a set X to a set Y . F is one-to-one (or injective) if, and only if, for all elements x1 and x2 in X,

• if F(x1) = F(x2), then x1 = x2,

• /equivalently, if x1 = x2, then F(x1) = F(x2).• Symbolically,– F: X → Y is one-to-one ⇔∀x1, x2 ∈ X, if F(x1) = F(x2) then x1 = x2.

On-to Functions

• Let F be a function from a set X to a set Y . • F is onto (or subjective) if, and only if, given any element y in

Y , it is possible to find an element x in X with the property – that y = F(x).

• Symbolically:– F: X → Y is onto ⇔∀y ∈ Y, ∃x ∈ X such that F(x) = y.

• A function is onto – if each element of the co-

domain has an arrow pointing to it from some element of the domain.

• A function is not onto – if at least one element in

its co-domain does not have an arrow pointing to it.