Set theory and Function

Course- BCA Subject- MATHEMATICS-I

Unit- I

RAI UNIVERSITY, AHMEDABAD

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Set-A collection of well defined objects (elements) is called a set. Notation There is a fairly simple notation for sets. You simply list each element, separated by a comma, and then put some curly brackets around the whole thing.

The curly brackets { } are sometimes called "set brackets" or "braces". This is the notation for the two previous

examples:

{Socks, shoes, watches, shirts,…} {Index, middle, ring, pinky}

The first set {socks, shoes, watches, shirts, ...} we call an infinite set, The second set {index, middle, ring, pinky} we call a finite set.

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters: {a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?) Numerical Sets So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers? Set of even numbers: {..., -4, -2, 0, 2, 4,...} Set of odd numbers: {..., -3, -1, 1, 3, ...} Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} Positive multiples of 3 that are less than 10: {3, 6, 9}and the list goes on. We can come up with all different types of sets. There can also be sets of numbers that have no common property; they are just defined that way. For example: {2, 3, 6, 828, 3839, 8827} {4, 5, 6, 10, 21} {2, 949, 48282, 42882959, 119484203}

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Universal Set—

At the start we used the word "things" in quotes. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to your question.

So far, all I've been giving you in sets are integers. So the universal set for all of this discussion could be said to be integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.

However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.

Some More Notations

When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set. So for example, A is a set, and a is an element in A. Same with B and b, and C and c.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in maths jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not? Also, when we say an element a is in a set A, we use the symbol to show it. And if something is not in a set use .

Example: Set A is {1, 2, 3}. You can see that 1 A, but 5 A Equality— Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely!

Example: Are A and B equal where: A is the set whose members are the first four positive whole numbers B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2, 3 and 4 and we have checked every element of both sets, so: Yes, they are!

and the equals sign (=) is used to show equality, so you would write: A = B

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:

A is a subset of B if and only if every element of A is in B. So let's use this definition in some examples.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}? 1 is in A, and 1 is in B as well. 3 is in A and 3 is also in B. 4 is in A, and 4 is in B. That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter; we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A? Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them. The sets are:

A = {..., -8, -4, 0, 4, 8, ...} B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:

So:

A is a subset of B, but B is not a subset of A

Proper Subsets— If we look at the definition of subsets and let our mind wander a bit, we come to a weird conclusion. Let A be a set. Is every element in A an element in A? (Yes, I wrote that correctly.)

Well, umm, yes of course, right? So wouldn't that mean that A is a subset of A? This doesn't seem very proper, does it? We want our subsets to be proper. So we introduce (what else but) 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. This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Example: {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

Example: {1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

You should notice that if A is a proper subset of B, then it is also a subset of B. Even More Notation:- When we say that A is a subset of B, we write A B. Or we can say that A is not a subset of B by A B ("A is not a subset of B") When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B. Empty (or Null) Set— This is probably the weirdest thing about sets. As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!" And right you are. It is a set with no elements. This is known as the Empty Set (or Null Set).There aren't any elements in it. It is represented by Or by {} (a set with no elements) Some other examples of the empty set are the set of countries south of the South Pole. So what's so weird about the empty set? Well, that part comes next.

Empty Set and Subsets— So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A? Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A. But what if we have no elements? It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true. A good way to think about it is: we can't find any elements in the empty set that aren't in A, so it must be that all elements in the empty set are in A. So the answer to the posed question is a resounding yes.

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

Order— 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. Just as there are finite and infinite sets, each has finite and infinite order. For finite sets, we represent the order by a number, the number of elements. Example— {10, 20, 30, 40} has an order of 4. For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinity is larger than others, but this is a more advanced topic in sets.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Definitions— • Denotes the empty set { }, which does not contain any elements. • N denotes the set of natural numbers {1, 2, 3 . . .}. • Z denotes the set of integers {. . . , −3, −2, −1, 0, 1, 2, 3 . . .}. • Q denotes the set of rational numbers {p/q: p, q ϵ Z with q≠ 0}. • R denotes the set of real numbers consisting of directed distances from a designated point zero on the continuum of the real line. • C denotes the set of complex numbers {a + bi: a, b ϵR with 푖 = √−1 } Z* = {. . . , −3, −2, −1, 1, 2, 3 . . .}, Z+ = {1, 2, 3 . . .}, Z- = {−1, −2, −3 . . .}. • A is a subset of B if every element of A is an element of B. We write A ⊆B and show A ∩⊆ B by proving that if a ∈ A, then a ∈B. • A is equal to B if A and B contain exactly the same elements. We write A = B and show A = B by proving both A ⊆ B and B ⊆ A. • A is a proper subset of B if A is a subset of B, but A is not equal to B. We write either A ⊂ B or A⊊ B and show A ⊂B by proving both A ⊆B and B ⊈A. Example: - let W = {1, 2}, X = {1, 3, 5}, and Y ={n : n is an odd integer}. We first prove X ⊆Y and then prove W ⊈ Y. Proof that X ⊆ Y We proves X ⊆ Y by showing that if a ϵ X, then a ϵ Y. Since X ={1, 3, 5} is finite, we prove this implication by exhaustion; that is, we consider every element of X one at a time and verify that each is in Y. Since 1 = 2 · 0 + 1, 3 = 2 · 1 + 1, and 5 = 2 · 2 + 1, each element of X is odd; in particular, each element of X has been expressed as 2k + 1 for some k ϵ Z). Thus, if a ϵ X, then a ϵ Y, and so X ⊆ Y. Venn Diagrams & Set Notation— The following examples should help you understand the notation, terminology, and concepts related to Venn diagrams and set notation. Let's say that our universe contains the numbers1, 2, 3, and 4. Let A be the set containing the numbers 1 and 2; that is, A = {1, 2}. (Warning: The curly braces are the customary notation for sets. Do not use parentheses or square brackets.) Let B be the set containing the numbers 2 and 3; that is, B = {2, 3}. Then we have the following relationships, with pinkish shading marking the solution "regions" in the Venn diagrams:

set notation pronunciation Meaning Venn diagram answer

A U B "A union B" everything that is in

either of the sets

{1, 2, 3}

A ^ B or

"A intersect B"

only the things that are in

both of the sets

{2}

Ac

or ~A

"A complement", or "not A"

everything in the universe

outside of A {3, 4}

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

A – B "A minus B", or "A complement B"

everything in A except for anything in its overlap with B

{1}

~(A U B) "not (A union B)" everything

outside A and B

{4}

~(A ^ B) or

~( ) "not (A intersect B)"

everything outside of the overlap

of A and B

{1, 3, 4}

There are gazillions of other possibilities for set combinations and relationships, but these are among the simplest and most common. Note that different texts use different set notation, so you should not be at all surprised if your text uses still other symbols than those used above. But while the notation may differ, the concepts will be the same. By the way, as you probably noticed, your Venn-diagram "circles" don't have to be perfectly round; ellipses will do just fine.

Given the following Venn diagram, shade in A ∩C.

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The intersection of A and C is just the overlap between those two circles, so:

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Given the following Venn diagram, shade in A U(B – C).

As usual when faced with parentheses, I'll work from the inside out.

I'll first find B – C. "B complement C" means I take B and then throw out its overlap with C, which gives me this:

Now I have to union this with A:

Note that union with A put some of C (that is, some of what I'd cut out when I did "B – C") back into the answer. This is okay. Just because we threw out C at one point, doesn't mean that it all has to stay out forever.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Function— Introduction—a function relates an input to an output.

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. For an example of a function, let X be the set consisting of four shapes-- a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors-- red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y-- each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function"

The input to a function is called the argument and the output is called thevalue. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the Co-domain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the Co-domain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

Definition— A function relates each element of a set with exactly one element of another set (possibly the same set). The Two Important Things—

1. "each element" means that every element in X is related to some element in Y. 2. "exactly one" means that a function is single valued.

(one-to-many) (many-to-one) This is NOT Correct in a function But this is Correct in a function

It is like a machine that has an input and an output. And the output is s related somehow to the input.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

If a relationship does not follow those two rules then it is not a function, it would still be a relationship, just not a function.

For Example— "Multiply by 2" is a very simple function

There is three parts—

Input Relationship Output

0 × 2 0

1 × 2 2

7 × 2 14

10 × 2 20

... ... ...

Exercise:- In the above relation for an input of 50, what would be the output? Some Example of Functions:-

풙ퟐ (Squaring) is a function. (풙ퟑ + ퟏ) is also a function. sine, cosine and tangents are the functions use in trigonometry. Name of the function—first, it is useful to give a name to a function. A most common name is "f", but we can give other names like ‘g’, ‘h’ etc.

But let's use "푓"-- We would say "푓 of x equals to x squared" So the function 푓(푥) = 1 − 푥 + 푥 is the same function if we write— 푓(푥) = 1 − 푥 + 푥 ℎ(푥) = 1 − 푥 + 푥 푤(푥) = 1− 푥 + 푥 It is just there so you know where to put the values--

푓(2) = 1 − 2 + 2 = 3

Also, 푓(2) = ℎ(2) = 푤(2)

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Relation—

At the top we discussed 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 you put into it! Saying "풇(ퟒ) = ퟏퟔ" is like saying 4 is somehow related to 16. or 4 → 16 Example— This tree grows 20 cm every year so the height of the tree is related to its age using the function h.

__________________________________________________________________________________________ Domain, Co-domain and Range— For a function 푓 − −푋 → 푌 The set "X" is called the Domain, The set "Y" is called the Co-domain, and The set of elements that get pointed to Y (the actual values produced by the function) is called the Range. Domain of 푓 is denoted by 퐷 . Range of 푓 is denoted by 푅 .

There are special name for what can be input to a function, and what will be output of that function—

(a) What can be input to a function is called the Domain. (b) What is the possible output of a function is called the Co-domain. (c) What is the actually output of a function is called the Range.

h(age)= age× 20 so if age is 10 year then height is h(10) = 10×20= 200cm If age is 15 Year then height of the tree is _________. (Try it!)

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

This is a function. You can tell by tracing from each x to each y. There is only one y for each x; there is only one arrow coming from each x.

Ha! Bet I fooled some of you on this one! This is a function! There is only one arrow coming from each x; there is only one y for each x. It just so happens that it's always the same y for each x, but it is only that one y. So this is a function; it's just an extremely boring function!

This one is not a function: there are two arrows coming from the number 1; the number 1 is associated with twodifferent range elements. So this is a relation, but it is not a function.

In this illustration— The set "A" is the Domain, The set "B" is the Co-domain, The set of elements that get pointed to in B (the

actual values produced by the function) are the Range, also called the Image.

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Okay, this one's a trick question. Each element of the domain that has a pair in the range is nicely well-behaved. But what about that 16? It is in the domain, but it has no range element that corresponds to it! This won't work! So then this is not a function. Heck, it isn’t even a relation!

1. Linear Function: A linear function is a function defined by an equation of the form 푦 = 푚푥 + 푐. 2. Quadratic Function: A quadratic function is a function defined by an equation of the form 푓(푥) = 푎푥2 + 푏푥 + 푐,

a ≠ 0. 3. Cubic Function: A function defined by a polynomial of degree 3. The general form of cubic function is

푓(푥) = 푎푥 + 푏푥 + 푐푥 + 푑, 푎 ≠ 0.

4. Polynomial: A polynomial is sum or difference of monomials, 푓(푥) =푎 푥 + 푎 푥 + 푎 푥 +⋯+ 푎 푥 + 푎 푥 + 푎

5. Rational Function: A function written in the form of ab is called rational function,

푓(푥) = 푝(푥)푞(푥), p(x) and q(x) are polynomial functions.

6. Implicit Function: An explicit function is one which is given in terms of the independent variable.

Take the following function, y = x2 + 3x - 8

y is the dependent variable and is given in terms of the independent variable x. Note that y is the subject of the formula.

7. Explicit Function: Implicit functions, on the other hand, are usually given in terms of both dependent and independent variables.

ex:- y + x2 - 3x + 8 = 0

Sometimes, it is not convenient to express a function explicitly. For example, the circle x2 + y2 = 16 could be written as 푦 = √16− 푥 or 푦 = −√16 − 푥

Unit-I Set theory and Function

RAI UNIVERSITY, AHMEDABAD

Reference— 1. en.wikipedia.org/wiki/Set_theory 2. plato.stanford.edu 3. www.math.toronto.edu 4. www.mathsisfun.com 5. www.mathgoodies.com 6. en.wikipedia.org/wiki/Function_(mathematics) 7. https://www.khanacademy.org 8. www.revolvy.com