math

1. Chenab College Shorkot
Khalid Jamil

2. Sets , function
Chenab College Shorkot
Khalid Jamil

4. set:
• A well-defined collection of distinct objects.
well-defined collection is meant a collection, which is
such that, given any object, we may be able to decide
whether the object belongs to the collection or not.
Distinct objects we mean objects no two of which are
identical (same)
The objects in a set are called its members or elements.
Capital letters A, B, C, X, Y, Z etc., are generally used as
names of sets
Small letters a, b, c, x, y, z etc., are used as members of
sets
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5. Chenab College Shorkot
Ways to describe sets
Descriptive method: Set may be described in words
Example : The set of all vowels of the English alphabets.
Tabular Method: A set may be described by listing its elements within brackets.
Example:A = {a,e,i,o,u}
Set-builder Method: using a symbol or letter for an arbitrary member of
the set and stating the property common to all the members.
A = { x : property }
Example: A = { x |x is a vowel of the English alphabet}

6. N = The set of all natural numbers = {1,2,3,...}
W = The set of all whole numbers = {0,1,2,...}
Z = The set of all integers = {0,±1,+2....}.
Z’ = The set of all negative integers = {-1,-2,-3,...}
O = The set of all odd integers = { ± 1,± 3,±5,...}.
E = The set of all even integers = {0,±2,±4,...}.
Q = The set of all rational numbers ={ I where and q 0}
Q’ = The set of all irrational numbers ={ I where and q 0}
= The set of all real numbers = Q Q’
∪
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7. Equal Sets: Two sets A and B are equal i.e.,
A=B, if and only if they have the same
elements that is, if and only if every element
of each set is an element of the other set.
Thus the sets { 1, 2, 3 } and { 2, 1, 3} are equal.
Note: (1) A = B if and only if they have the
same elements means if A = B they have the
same elements and if A and B have the same
elements then A = B.
(2) The phrase if and only if is shortly written as
“iff”.
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8. Equivalent Sets: If the elements of two sets A and B
can be paired in such a way that each element of
A is paired with one and only one element of B
and vice versa, then such a pairing is called a
one-to-one correspondence between A and B
e.g., if A = {Billal, Ahsan, Jehanzeb} and B =
{Fatima, Ummara, Samina} then six different (1 - 1)
correspondences can be established between A
and B Two of these correspondences are shown
below;
symbol ~ is used to mean is equivalent to
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9. e.g
Order of a Set: The number of members/elements of a set.
A set having only one element is called a singleton set .
Set with no element (zero number of elements) is called the empty set
or null set.
The empty set is denoted by the symbol φ or { }.
Finite and In-Finite sets: If a set is equivalent to the set {1, 2, 3,...n}
for some fixed natural number n, then the set is said to be infinite
otherwise in-finite. Chenab College Shorkot

10. Subset:: If every element of a set A is an element of set
B, then A is a subset of B.
Symbolically this is written as: A B
⊆ (A is subset of B)
B is a super set of A. written as: B A
⊇ {B is a superset
of A}
Proper Subset: If A is a subset of B and B contains at
least one element which is not an element of A, then A
is said to be a proper subset of B.
A B (A is a proper subset of B).
⊂
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11. Improper Subset: If A is subset of B and A = B, then we say
that A is an improper subset of B.
It also follows that every set A is an improper subset of itself.
• The empty set is a subset of every set.
Power Set: The set containing all the possible subsets of a set
power set of a set S denoted by P (S).
Example: If A = {a, b}, then P(A)= { {}, {a} , {b} ,{a, b} }
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12. universal set: A set which contains all the sets relevant to a
certain condition is called the universal set. It is the set of all
possible values.
Example: If A = {1,2,3} and B {2,3,4,5}, then universal set
here will be:
U = {1,2,3,4,5}
Note: (1) The power set of the empty set is not empty.
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