A set is a group or collection of objects or numbers, considered as an entity
unto itself. Sets are usually symbolized by uppercase, italicized, boldface
letters such as A, B, S, or Z. Each object or number in a set is called a
member or element of the set. Examples include the set of all computers in
the world, the set of all apples on a tree, and the set of all irrational numbers
between 0 and 1.
A set is a collection of things, usually numbers. We can list each element (or
"member") of a set inside curly brackets.
In mathematics, a set is a collection of well defined distinct objects, considered as an object in its
own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but
when they are considered collectively they form a single set of size three, written {2, 4, 6}. The
concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th
century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from
which nearly all of mathematics can be derived. In mathematics education, elementary topics from
set theory such as Venn diagrams are taught at a young age, while more advanced concepts are
taught as part of a university degree.
Common Symbols Used in Set Theory
Symbols save time and space when writing. Here are the most common set
symbols
In the examples C = {1,2,3,4} and D = {3,4,5}
Symbo
Meaning Example
l
{} Set: a collection of elements {1,2,3,4}
A ∪ B Union: in A or B (or both) C ∪ D = {1,2,3,4,5}
A ∩ B Intersection: in both A and B C ∩ D = {3,4}
A ⊆ B Subset: A has some (or all) elements of B {3,4,5} ⊆ D
A ⊂ B Proper Subset: A has some elements of B {3,5} ⊂ D
A ⊄ B Not a Subset: A is not a subset of B {1,6} ⊄ C
A ⊇ B Superset: A has same elements as B, or more {1,2,3} ⊇ {1,2,3}
A ⊃ B Proper Superset: A has B's elements and {1,2,3,4} ⊃ {1,2,3}
more
A ⊅ B Not a Superset: A is not a superset of B {1,2,6} ⊅ {1,9}
Dc = {1,2,6,7}
Ac Complement: elements not in A
When = {1,2,3,4,5,6,7}
A−B Difference: in A but not in B {1,2,3,4} − {3,4} = {1,2}
a ∈ A Element of: a is in A 3 ∈ {1,2,3,4}
b ∉ A Not element of: b is not in A 6 ∉ {1,2,3,4}
∅ Empty set = {} {1,2} ∩ {3,4} = Ø
Universal Set: set of all possible values
(in the area of interest)
P(A) Power Set: all subsets of A P({1,2}) = { {}, {1}, {2}, {1,2} }
A=B Equality: both sets have the same members {3,4,5} = {5,3,4}
Cartesian Product {1,2} × {3,4}
A×B
(set of ordered pairs from A and B) = {(1,3), (1,4), (2,3), (2,4)}
|A| Cardinality: the number of elements of set A |{3,4}| = 2
| Such that { n | n > 0 } = {1,2,3,...}
: Such that { n : n > 0 } = {1,2,3,...}
∀ For All ∀x>1, x2>x
∃ There Exists ∃ x | x2>x
∴ Therefore a=b ∴ b=a
Natural Numbers {1,2,3,...} or {0,1,2,3,...}
Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers
Algebraic Numbers
Real Numbers
Imaginary Numbers 3i
Complex Numbers 2 + 5i
There are two ways of describing, or specifying the members of, a set. One way is by intensional
definition, using a rule or semantic description:
A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.
The second way is by extension – that is, listing each member of the set. An extensional
definition is denoted by enclosing the list of members in curly brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}.
One often has the choice of specifying a set either intensionally or extensionally. In
the examples above, for instance, A = C and B = D.
In an extensional definition, listing a member repeatedly does not change the set,
for example, the set {11, 6, 6} is identical to the set {11, 6}. Moreover, the order in
which the elements of a set are listed is irrelevant (unlike for a sequence or tuple),
so {6, 11} is yet again the same set.
For sets with many elements, the enumeration of members can be abbreviated. For
instance, the set of the first thousand positive integers may be specified
extensionally as
{1, 2, 3, ..., 1000},
where the ellipsis ("...") indicates that the list continues in the obvious way.
The notation with braces may also be used in an intensional specification of a
set. In this usage, the braces have the meaning "the set of all ...". So, E =
{playing card suits} is the set whose four members are spades, diamonds,
hearts, and clubs. A more general form of this is set-builder notation, through
which, for instance, the set F of the twenty smallest integers that are four less
than a perfect square can be denoted
In this notation, the colon (":") means "such that", and the description can
be interpreted as "F is the set of all numbers of the form n2 − 4, such
that n is an integer in the range from 0 to 19 inclusive". Sometimes
the vertical bar ("|") is used instead of the colon.
Lesson on Subsets
Example 1: Given A = {1, 2, 4} and B = {1, 2, 3, 4, 5},
what is the relationship between these sets?
We say that A is a subset of B, since every element of A is also in B. This is denoted
by:
A Venn diagram for the relationship between these sets is shown to the right.
Answer: A is a subset of B.
Proper subset definition
A proper subset of a set AA is a subset of AA that is not equal to AA. In other words, if BB is a proper
subset of AA, then all elements of BB are in AA but AA contains at least one element that is not
in BB.
For example, if A={1,3,5}A={1,3,5} then B={1,5}B={1,5} is a proper subset of AA. The
set C={1,3,5}C={1,3,5} is a subset of AA, but it is not a proper subset of AA since C=AC=A. The
set D={1,4}D={1,4} is not even a subset of AA, since 4 is not an element of AA.
There are two basic differences between a proper and an improper subset. The first
one is that no proper subset is same as the given set while the improper subset is. The
second one is that every set has only one improper subset while it can have 1 or more
than 1 improper subsets.
It is important to note that the sets that consist of only one element have no proper
subset other than NULL and as always exactly one improper subset which is the set
itself. In cases where we have empty sets, there is even no proper subset but an
improper subset still exists. This implies no matter what, improper subset of a set
always exists whether or not it has a proper subset of it.
All The Subsets
For the set {a,b,c}:
The empty set {} is a subset of {a,b,c}
And these are subsets: {a}, {b} and {c}
And these are also subsets: {a,b}, {a,c} and {b,c}
And {a,b,c} is a subset of {a,b,c}
And altogether we get the Power Set of {a,b,c}:
P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
Think of it as all the different ways we can select the items (the order of the
items doesn't matter), including selecting none, or all.
1.2.3 Cardinality: Countable and Uncountable
Sets
Here we need to talk about cardinality of a set, which is basically the size
of the set. The cardinality of a set is denoted by |A||A|. We first discuss
cardinality for finite sets and then talk about infinite sets.
Finite Sets:
Consider a set AA. If AA has only a finite number of elements, its cardinality
is simply the number of elements in AA. For example,
if A={2,4,6,8,10}A={2,4,6,8,10}, then |A|=5|A|=5. Before discussing
infinite sets, which is the main discussion of this section, we would like to
talk about a very useful rule: the inclusion-exclusion principle. For two
finite sets AA and BB, we have
|A∪B|=|A|+|B|−|A∩B|.|A∪B|=|A|+|B|−|A∩B|.
To see this, note that when we add |A||A| and |B||B|, we are counting the elements in |A∩B||
A∩B| twice, thus by subtracting it from |A|+|B||A|+|B|, we obtain the number of elements in |
A∪B||A∪B|, (you can refer to Figure 1.16 in Problem 2 to see this pictorially). We can extend the
same idea to three or more sets.
1.2.2 Set Operations
The union of two sets is a set containing all elements that are
in AA or in BB (possibly both). For example, {1,2}∪{2,3}={1,2,3}
{1,2}∪{2,3}={1,2,3}. Thus, we can write x∈(A∪B)x∈(A∪B) if and only
if (x∈A)(x∈A) or (x∈B)(x∈B). Note that A∪B=B∪AA∪B=B∪A. In Figure
1.4, the union of sets AA and BB is shown by the shaded area in the Venn
diagram.
Fig.1.4 - The shaded area shows the set B∪AB∪A.
14.4 Union and intersection (EMA7Z)
Union
The union of two sets is a new set that contains all of the elements that are in at
least one of the two sets. The union is written as A∪B or “A or B”.
Intersection
The intersection of two sets is a new set that contains all of the elements that are
in both sets. The intersection is written as A∩B or “A and B”.
The figure below shows the union and intersection for different configurations of two
events in a sample space, using Venn diagrams.
Figure 14.1: The unions and intersections of different events. Note that in the middle
column the intersection, A∩B, is empty since the two sets do not overlap. In the final
column the union, A∪B, is equal to A and the intersection, A∩B, is equal to B since B is
fully contained in A.
UNION, INTERSECTION, AND COMPLEMENT
The union of two sets contains all the elements contained in either set (or both sets).
The union is notated A ⋃ B.
More formally, x ∊ A ⋃ B if x ∊ A or x ∊ B (or both)
The intersection of two sets contains only the elements that are in both sets.
The intersection is notated A ⋂ B.
More formally, x ∊ A ⋂ B if x ∊ A and x ∊ B
The complement of a set A contains everything that is not in the set A.
The complement is notated A’, or Ac, or sometimes ~A.
What is Number Line?
In math, a number line can be defined as a straight line with numbers placed at
equal intervals or segments along its length.
A number line can be extended infinitely in any direction and is usually represented
horizontally.
The numbers on the number line increase as one moves from left to right and decrease
on moving from right to left.
from right to left.
Here, fractions and decimals have been represented on the number line.
A number line can also be used to represent both positive and negative integers as
well.
Writing numbers on a number line makes comparing numbers easier. Numbers on the
left are smaller than the numbers on the right of the number line.
A number line can also be used to carry out addition, subtraction and multiplication. We
always move right to add, move left to subtract and skip count to multiply.
Fun Facts
A blank number line was originally proposed as a visual model or diagram for solving addition
and subtraction operations. An empty or blank number line is a visual diagram of a number line
with no numbers or markers and is essentially used for solving word problems.
