Sets and Functions

By:Muhammad Adnan EjazSets, Functions and Groups

Definition

Inmathematics, asetis 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 singlesetof size three, written {2,4,6}.

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:

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

Subsets

When we define a set, if we take pieces of that set, we can form what is called asubset.

So for example, we have the set {1, 2, 3, 4, 5}. Asubsetof 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:

Cont

A is a subset of B if and only if every element of A is in B.

Proper Subsets

A is apropersubset of B if and only if every element in A is also in B, and there existsat least one elementin B that isnotin A.

{1, 2, 3}isaproper subsetof {1, 2, 3, 4} because the element 4 is not in the first set.

Empty (or Null) Set

It is a set withno elements.It is represented by{}(a set with no elements)

The empty set is a subset of every set, including the empty set itself.

Order

No, not the order of the elements. In sets itdoes 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 meanthe size of the set.Example, {10, 20, 30, 40} has an order of 4

Functions

Inmathematics, afunctionis 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 functionthat relates each real number x to its square x2.A function relates an input to an output.

Classic way of writing

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

Examples

x2(squaring) is a functionx3+1is a functionSine, Cosine and Tangentare functions used in trigonometry

Function Name

First, it is useful to give a function aname.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

Example

withf(x) = x2:an input of 4becomes an output of 16.In fact we can writef(4) = 16.

Relating

At the top we said that a function waslikea 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 functionrelatesan input to an output.

Example

Example: this tree grows 20 cm every year, so the height of the tree isrelatedto its age using the functionh:h(age) = age 20So, if the age is 10 years, the height is:h(10) = 10 20 = 200 cm

Rules

A function hasspecial rules:It must work foreverypossible input valueAnd it has onlyone relationshipfor each input valueA function relateseach elementof a setwithexactly oneelement of another set(possibly the same set).

Cont

"...each element..."means that every element inXis related to some element inY.We say that the functioncoversX(relates every element of it).(But some elements ofYmight not be related to at all, which is fine.)"...exactly one..."means that a function issingle valued. It will not give back 2 or more results for the same input.So "f(2) = 7or9" is not right!

Explicit vs. Implicit

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

Cont

"Implicit" is when it isnotgiven directly such as:x2- 3xy + y3= 0When 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 shownindirectly.

Groups21Definition of Groups A group (G, ) is a set G together with a binary operation satisfying the following axioms.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 a1 G such that a-1 a = a a1 = e.

Cont22Examples of GroupsExample 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 11 = 1, (1)1 = 1, i1 = i, and (i)1 = i.

Cont23SubgroupsIt often happens that some subset of a group willalso form a group under the same operation.Sucha group is called a subgroup. If (G, ) is agroup and H is a nonempty subset of G, then (H, ) is called a subgroup of (G, ) if thefollowing conditions hold:(i) a b H for all a, b H. (closure)(ii) a1 H for all a H. (existence of inverses)

Cont24If 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 themathematician Niels Abel.

Cont25Conditions (i) and (ii) are equivalent to the single condition: (iii) a b1 H for all a, b H.

Cont26Definition. If G is a group and a G, write = {an : n Z} = {all powers of a } . It is easy to see that 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.

Cont27

Definition. If G is a finite group, then the number of elements in G, denoted by G, is called the order of G.

Theorem28Theorem Lagranges 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 unionG = 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|.

Normal Subgroups29Normal SubgropsLet 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 ab1 H. We can also define the relation L on G so that aLb if and only if b1a 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.

Subgroup30Definition: A subgroup H of a group G is called a normal subgroup of G if g1hg H for all g G and h H.