This document discusses sets, set operations, Cartesian products, and functions. It defines what sets and functions are, how they are represented, and different types of sets and functions. Sets are collections of distinct objects that can be determined if an object is a member. Set operations include union, intersection, and difference. Cartesian products combine sets of ordered pairs. Functions map inputs to outputs and can be one-to-one, onto, into, or bijective based on whether elements are mapped uniquely.

1. Prepared by: Daniyal Khan
MCS 2nd Semester
NCBA&E RYK
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2. We will be discussing
 Sets
 Set Notations
 Set operations
 Cartesian Product
 Functions
 Types of Functions
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3. Sets are used to define the concepts of relations and functions.
The study of geometry, sequences, probability, etc. requires the knowledge of sets.
Studying sets helps us categorize information.
It allows us to make sense of a large amount of information by breaking it down
into smaller groups.
We check for collection!
Collection is not necessary
If collection exist.. Must be distinct( different, unique )
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4.  Well defined collection of distinct objects!
 Definition: A set is any collection of objects specified in such a way that we can
determine whether a given object is or is not in the collection.
 In other words A set is a collection of objects.
 These objects are called elements or members of the set. The following points are
noted while writing a set.
 Sets are usually denoted by capital letters A, B, S, etc. The elements of a set are
usually denoted by small letters a, b, t, u, etc .
 Examples: A = {a, b, d, 2, 4} B = {math, religion, literature, computer science }
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5.  By convention, particular symbols are reserved for the most important sets of
numbers:
 ∅ – empty set .
 C – complex numbers .
 N – natural numbers .
 Q – rational numbers (from quotient)
 R – real numbers
 Z – integers(from Zahl, German for integer).
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6.  Empty Sets: A set that contains no members is called the empty set or null set .
The empty set is written as { } or ∅.
 Finite Sets: A set is finite if it consists of a definite number of different elements.
If W be the set of people living in a town, then W is finite.
 Infinite Sets: An infinite set is a set that is not a finite set. Infinite sets may be
countable or uncountable. The set of all integers, {..., -1, 0, 1, 2, ...}, is a count ably
infinite set;
 Equal Sets: Equal sets are sets which have the same members. If P ={1,2,3},
Q={2,1,3}, R={3,2,1} then P=Q=R.
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7.  Subsets: Sets which are the part of another set are called subsets of the original
set. For example, if A={1,2,3,4} and B ={1,2} then B is a subset of A it is
represented by ⊆.
 Power Sets: If ‘A’ is any set then one set of all are subset of set ‘A’ that it is called a
power set. Example- If S is the set {x, y, z}, the power set of S is {{}, {x}, {y}, {z}, {x,
y}, {x, z}, {y, z}, {x, y, z}}.
 Universal Sets: A universal set is a set which contains all objects, including itself.
Example- A={12345678} B={1357} C={2468} D={2367} Here A is universal set and
is denoted by U.
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8.  Union of sets
 Intersection of sets
 Difference of sets
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9.  The union of two sets would be wrote as A U B, which is the set of elements that
are members of A or B, or both too
 Intersection are written as A ∩ B, is the set of elements that are in A and B.
 If A is any set which is the subset of a given universal set then its complement is
the set which contains all the elements that are in but not in A.
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10.  Venn diagrams are named after a English logician, John Venn.
 It is a method of visualizing sets using various shapes.
 These diagrams consist of rectangles and circles.
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11.  AxB
 Is the set of all possible ordered pairs with the first element of each pair taken from A
and the second element from B
 If A is the set {1,2} and B is the set {x,y,z}
AxB = {(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)}
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12.  A function is an activity
 Also written as f(x)
 A is the domain of f
 B is the range of f
 x is the input or argument
 f(x) is the output or image
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f: A B

13.  Into function
 Onto function
 One to one function
 Bijective function
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14.  A relation R is said to be into if R != B
 B is set of second elements in a relation
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15.  A function is onto if every element of its range is the image of some element of its
domain.
 If R = B
 2nd element may or may not repeat
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16.  A function is one-to-one if no two elements of its domain are mapped into the same
element of its range.
 If 2nd element doesn’t repeat
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17.  A function that is both one-to-one and onto
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18. We have discussed
 Sets
 Set Notations
 Set operations
 Cartesian Product
 Functions
 Types of Functions
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