Mathematics Realted sets, functions and sets

1. By:
Muhammad Adnan Ejaz
Sets, Functions and Groups

2. Definition
 In mathematics, a set is a collection of
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}.

3. Notation
 There is a fairly simple notation for sets. We
simply list each element (or "member")
separated by a comma, and then put some
curly brackets around the whole thing:

4. Why are sets?
 Sets are the fundamental property of
mathematics. Now as a word of warning, sets,
by themselves, seem pretty pointless. But it's
only when we apply sets in different situations
do they become the powerful building block of
mathematics

5. Subsets
 When we define a set, if we take pieces of
that set, we can form what is called a subset.
 So for example, we have the set {1, 2, 3, 4, 5}.
A subset of this is {1, 2, 3}. Another subset is
{3, 4} or even another, {1}. However, {1, 6} is
not a subset, since it contains an element (6)
which is not in the parent set. In general:

6. Cont…
 A is a subset of B if and only if every element of A
is in B.

7. Proper Subsets
 A is a proper subset of B if and only if every
element in A is also in B, and there exists at
least one element in B that is not in A.
 {1, 2, 3} is a proper subset of {1, 2, 3, 4}
because the element 4 is not in the first set.

8. Empty (or Null) Set
 It is a set with no elements.It is represented
by {} (a set with no elements)
 The empty set is a subset of every set,
including the empty set itself.

9. Order
 No, not the order of the elements. In sets
it does not matter what order the elements
are in.
 Example: {1,2,3,4} is the same set as {3,1,4,2}
 When we say "order" in sets we mean the
size of the set.
 Example, {10, 20, 30, 40} has an order of 4

10. Functions
 In mathematics, a function is a relation
between a set of inputs and a set of
permissible outputs with the property that
each input is related to exactly one output. An
example is the function that relates each real
number x to its square x2.
 A function relates an input to an output.

11. Classic way of writing
 "f(x) = ... " is the classic way of writing a function.

12. Examples
 x2 (squaring) is a function
 x3+1 is a function
 Sine, Cosine and Tangent are functions used in
trigonometry

13. Function Name
 First, it is useful to give a function a name.
 The most common name is "f", but we can
have other names like "g" ... or even
"marmalade" if we want.
 We say "f of x equals x squared“

14. Example
 with f(x) = x2:
 an input of 4
 becomes an output of 16.
 In fact we can write f(4) = 16.

15. Relating
 At the top we said that a function was like a
machine. But a function doesn't really have
belts or cogs or any moving parts - and it
doesn't actually destroy what we put into it!
 A function relates an input to an output.

16. Example
 Example: this tree grows 20 cm every year, so
the height of the tree is related to its age
using the function h:
 h(age) = age × 20
 So, if the age is 10 years, the height is:
 h(10) = 10 × 20 = 200 cm

17. Rules
 A function has special rules:
 It must work for every possible input value
 And it has only one relationship for each
input value
 A function relates each element of a set
with exactly one element of another set
(possibly the same set).

18. Cont…
 "...each element..." means that every
element in X is related to some element
in Y.
 We say that the
function covers X (relates every element
of it).
 (But some elements of Y might not be
related to at all, which is fine.)
 "...exactly one..." means that a function
is single valued. It will not give back 2 or
more results for the same input.
 So "f(2) = 7 or 9" is not right!

19. Explicit vs. Implicit
 "Explicit" is when the function shows us how
to go directly from x to y, such as:
 y = x3 - 3
 When we know x, we can find y
 That is the classic y = f(x) style.

20. Cont…
 "Implicit" is when it is not given directly such
as:
 x2 - 3xy + y3 = 0
 When we know x, how do we find y?
 It may be hard (or impossible!) to go directly
from x to y.
 "Implicit" comes from "implied", in other words
shown indirectly.

21. Groups
21
Definition of Groups
A group (G, ・) is a set G together with a binary
operation ・ satisfying the following axioms.
(i) The operation ・ is associative; that is,
(a ・ b) ・ c = a ・ (b ・ c) for all a, b, c ∈ G.
(ii) There is an identity element e ∈ G such that
e ・ a = a ・ e = a for all a ∈ G.
(iii) Each element a ∈ G has an inverse element a−1
∈ G such that a-1・ a = a ・ a−1 = e.

22. Cont…
22
Examples of Groups
 Example Let G be the set of complex numbers
{1,−1, i,−i} and let ・ be the standard multiplication of
complex numbers. Then (G, ・) is an abelian group.
The product of any two of these elements is an
element of G; thus G is closed under the operation.
Multiplication is associative and commutative in G
because multiplication of complex numbers is always
associative and commutative. The identity element is
1, and the inverse of each element a is the element
1/a. Hence
1−1 = 1, (−1)−1 = −1, i−1 = −i, and (−i)−1 = i.

23. Cont…
23
 Subgroups
It often happens that some subset of a group
will
also form a group under the same
operation.Such
a group is called a subgroup. If (G, ・) is a
group and H is a nonempty subset of G, then
(H, ・) is called a subgroup of (G, ・) if the
following conditions hold:
(i) a ・ b ∈ H for all a, b ∈ H. (closure)
(ii) a−1 ∈ H for all a ∈ H. (existence of inverses)

24. Cont…
24
If the operation is commutative, that is,
if a ・ b = b ・ a for all a, b ∈ G,
the group is called commutative or abelian, in
honor of the
mathematician Niels Abel.

25. Cont…
25
 Conditions (i) and (ii) are equivalent to the single
condition:
(iii) a ・ b−1 ∈ H for all a, b ∈ H.

26. Cont…
26
 Definition. If G is a group and a  G, write
<a > = {an : n Z} = {all powers of a } .
It is easy to see that <a > is a subgroup of G .
< a > is called the cyclic subgroup of G
generated by a. A group G is called cyclic if there
is some a  G with G = < a >; in this case a is
called a generator of G.

27. Cont…
27
 Definition. If G is a finite group, then the number
of elements in G, denoted by G, is called the
order of G.

28. Theorem
28
 Theorem Lagrange’s Theorem. If G is a finite
group and H is a subgroup of G, then |H| divides
|G|.
Proof. The right cosets of H in G form a partition of
G, so G can be written as a disjoint union
G = Ha1 ∪ Ha2 ∪ ·· ·∪ Hak for a finite set of
elements a1, a2, . . . , ak ∈ G.
By Lemma 2.2.1, the number of elements in each
coset is |H|. Hence, counting all the elements in
the disjoint union above, we see that |G| = k|H|.
Therefore, |H| divides |G|.

29. Normal Subgroups
29
 Normal Subgrops
 Let G be a group with subgroup H. The right
cosets of H in G are equivalence classes under
the relation a ≡ b mod H, defined by ab−1 ∈ H. We
can also define the relation L on G so that aLb if
and only if b−1a ∈ H. This relation, L, is an
equivalence relation, and the equivalence class
containing a is the left coset aH = {ah|h ∈ H}. As
the following example shows, the left coset of an
element does not necessarily equal the right
coset.

30. Subgroup
30
Definition: A subgroup H of a group G is called a
normal subgroup of G if g−1hg ∈ H for all g ∈ G
and h ∈ H.