The Four Basic Concepts of Mathematics

There are four basic concepts in mathematics: sets, relations, functions, and binary operations. They are called "basic concepts" because these four are present in any field of mathematics. Mastery of these four can lead to better understanding of future mathematical topics.

In this lecture, we will define fundamental terms under these concepts and then learn how to apply them in writing and speaking mathematically.

The study of sets is called Set Theory [1]. The concept of sets is used in almost all areas of math including algebra, calculus, probability, and geometry. A set is a well-defined collection of objects. By well-defined, we mean that membership to the collection is precise, unambiguous, and with distinguishable limits. For example, "the collection of all vowels in the English alphabet" is a set but, the "collection of all beautiful faces" is not a set because "beautiful faces" is not a well-defined characteristic.

Any member of a set is called an element. For example, since "a" is a vowel, then "a" is an element of the set mentioned above. If a is an element of S, then we write a ∈ S. Otherwise, if a is not an element of S, then we write a ∉ S. Sets are usually denoted with an English capital letter such as "S".

Some standard sets in Math are:

• Set of natural numbers, ℕ = {1, 2, 3, ...}

• Set of whole numbers, W = {0, 1, 2, 3, ...}

• Set of integers, ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Set of rational numbers, ℚ = {p/q | q is an integer and q ≠ 0}

• Set of irrational numbers, ℚ' = {x | x is not rational}

• Set of real numbers, ℝ = ℚ ∪ ℚ'

All these are infinite sets. But there can be finite sets as well. For example, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set. 

The cardinal number, cardinality, or order of a set denotes the total number of elements in the set. Cardinality of a set refers to the total number of elements present in a set. It describes the size of a set.

Example 1: In the set A = {2, 3, 4, 6, 7}, there are 5 elements. Thus, the cardinality is 5.

Example 2: The cardinality of the set  A = {1, 5, 3, 2, 10, 6, 4}, is 7 because the set has 7 elements. 

Cardinality Symbols

The cardinality of a set X is denoted by |X|. 

We can also represent the cardinality of the set X as n(X).

For example, given set A = {1, 2, 3}. Then | A | = 3 or given set B = {a, b, c, 1, 4, 5, 8}. Then, n(B) = 7.

There are two ways of writing out sets:

1. Roster Notation. In roster notation, the elements of the set are listed.

S = {a, e, i, o, u}

2. Set Builder Notation. In the set builder notation, the characteristic that defines the collection is specified.

S = {x | x is a vowel in the English alphabet}

Here, x functions as a variable (and not the actual letter x). This is read as "S is the set of all objects x such that x is a vowel in the English alphabet."

1. Semantic Form. Semantic notation describes a statement to show what are the elements of a set. 

For example, a set of the first five odd numbers.

2. Visual Representation of Sets Using Venn Diagram. Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles. Sometimes a rectangle encloses the circles, which represents the universal set. The Venn diagram represents how the given sets are related to each other.

Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.

Singleton Sets

• A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}.

Finite Sets

• As the name implies, a set with a finite or countable number of elements is called a finite set. Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19} 

Infinite Sets

• A set with an infinite number of elements is called an infinite set. Example: Set C = {Multiples of 3}.

Empty or Null Sets

• A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = { }.

Equal Sets

• Two sets are said to be equal if they have exactly the same elements. That is, A = B if and only if every element of A is also an element of B and at the same time, every element of B is also an element of A. For example,

A = {a, e, i, o, u} and B = {x | x is a vowel in the English alphabet},

then A = B

Unequal Sets

• If two sets have at least one different element, then they are unequal sets. Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B.

Equivalent Sets

• Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

Overlapping Sets

• Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.

Disjoints Sets

• Two sets are disjoint if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.

Subset and Superset

Universal Set

• There is a set which contains all elements of interest. We call this the universal set denoted by U. Since it contains all elements, any set is a subset of the universal set. There is also a set which does not contain any element. This is called the empty set or null set which is denoted by either {} or ∅. It can also be shown that the empty set is a subset of any set. 

Power Sets

• Power set is the set of all subsets that a set could contain. The power set is denoted by the notation P(S) and the number of elements of the power set is given by 2^n. Example: Set A = {1,2,3}. Power set of A, P(A) = {∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}} = 8 or 2^3 = 8.

Other Important things to know about set: 

• The way the elements are arranged in the roster notation does not matter. That is,

{a, e, i, o, u} = {u, o, i, e, a} = {e, o, a, u, i}

• Each element in the roster notation is said to be unique. This is the same as saying that if the same element is written twice, then both are counted as one.

{a, e, i, o, u} = {a, e, i, o, u, a}

• A set cannot be an element of itself.

Like numbers, there are also operations for sets. These are:

• union (∪);

• intersection (∩);

• universal complement (ᶜ); and,

• relative complement (-).

• cartesian product of sets (S x S)

For example, if U = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} and A = {a, b, c, d}, then, 

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Then, A - B = {1, 2}. 

For example, A = {1, 2} and B = {1, 3}, then:

• A x B = {(1, 1), (1, 3), (2, 1), (2, 3)}.

• B x A = {(1, 1), (1, 2), (3, 1), (3, 2)}

• A x A =  A² = {(1, 1), (1, 2), (2, 1), (2, 2)}

• B x B = B²  = {(1, 1), (1, 3), (3, 1), (3, 3)}

Given two non-empty sets A and B, their Cartesian product A × B is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B. Mathematically, 

Reference(s)

[1] J. Bagaria. Set Theory. Stanford Encyclopedia of Philosophy. 2019. From https://plato.stanford.edu/entries/set-theory/

Can you name all of the following symbols? 

These symbols are called relations. Their job is to establish a connection between two numbers. When any of these symbols are found between two numbers, it means that the two numbers satisfy a certain condition required for such relation. For example, when we write 

it means that 1.999 and 2 satisfy the relation of being "approximately equal" to each other. Note that relations are different from operations like plus (+), minus (-), times (×), and divide (÷). Operations generate a certain result while operations are only statements which may be identified as either true or false.

In this part of Lecture 06, we will understand what relations are in the theoretical level and how they are used to aid mathematical work.

Question. Is A × B = B × A? 

It is also possible to have a Cartesian product of a set with itself. For example, 

For faster writing we sometimes denote these by 

In some cases, triple (and even multiple) products are defined like A × B × C, and 

However for this class, we will not be dealing with multiple Cartesian products. Instead we will restrict our coverage with pairs. 

Cartesian products are used in mathematics to construct a set out of other sets that are useful for some purpose. For example, the Cartesian plane which is a set of ordered pairs (x, y) where x and y are both real numbers is simply the Cartesian product 

In probability, Cartesian products express the sample space of experiments. For example, if a man has two t-shirts of colors S = {red, green} and two pants of colors P = {black, blue}. Then all the possible outfits he could wear are given by the Cartesian product:

S × P = {(red, black), (red, blue), (green, black), (green, blue)}

Given a set A, we define a relation on A as a subset of A². For now, we will denote relations by R. For example, if A = {1, 2, 3}, then A² = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}. Thus, any of the following can be considered as relations on A. 

If R is a relation on A and (a,b) ∈ R, then we can write aRb (read as a is in Relation to b). Hence, from above, we can say 

Exercise 4. Let A = {a, b, c}. Let ~ (called "tilde", read as "TIL-duh") be a relation on A defined by

~ = {(a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,c)}

Note: From this point forward, we will mostly use symbols like ~ for relations instead of the letter R. 

Some relations are more important than others because they satisfy certain special properties. Some kinds of relations that we are interested in are reflexive, symmetric, and transitive relations. 

A relation on a set A is reflexive if and only if every element of A is related to itself. In other words, 

For example, if A = {1, 2, 3}, then 

A relation on a set A is symmetric if and only if a is related to b implies b is related to a. In other words, 

For example, if A = {1, 2, 3}, then 

A relation on a set A is transitive if and only if a is related to b and b is related to c implies a is related to c. In other words, 

For example, if A = {1, 2, 3}, then 

A relation is transitive only when if aRb and bRc exists then we need to check for aRc. If it does not exists, there is no point of discussing about aRc with out bRc. Which implies that there is no counter example to disprove, hence transitive. 

EXERCISE NO 6. 

Consider a relation ^ on A = {1, 2, 3, 4} defined as ^ = {(a, b) | a divides b}; a, b ∈ A. Write ^ as a set of ordered pairs and prove whether ^ is:

(i) reflexive;

(ii) symmetric; and

(iii) transitive?

EXERCISE NO. 7

Consider a relation # on A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} defined as # = {(a, b) | b = a + 1}; a, b ∈ A. Write # as set of ordered pairs and prove if # is:

(i) reflexive?

(ii) symmetric; and

(iii) transitive?

EXERCISE NO. 8. 

Write the following Relation as a set of ordered pairs and prove whether the given relation is (i) Reflexive, (ii) Symmetric, and (iii) transitive. 

1. Consider a relation % on A = {1, 2, 3, 4, ... , 12, 13, 14} defined as % = {(x, y) | 3x - y = 0}; x, y ∈ A. 

2. Consider a relation # on the set of Natural Numbers, N defined as # = {(x, y) | y = x + 5 and x < 10}; x, y ∈ N.

We will now look at a special type of relation - functions. According to many mathematicians including Derbyshire[1], functions are the second or third most important objects in mathematics after numbers and sets. We will try to answer the question: What makes a function? And then, we will look at a few examples and their applications in the modern world. 

Definition of a Function

It is best to imagine a function to be like a machine because they work in exactly the same principle. We feed something into the machine, we call this the input, then the machine processes it in order to produce what we call the output. 

If we denote the function by f, the input by x, and the output by y, then we write 

There are conventional mathematical terms for inputs and outputs in a function. The input, x, is called the pre-image while the output, y, is called the image. We read the notation as "f of x is y" or "f maps x to y". In fact, in other branches of mathematics, functions are interpreted as mappings. 

For this example, we have 

A function maps the set of pre-images (called the domain) into a set where the images are found (called the codomain). In some cases, not all elements in the codomain have a pre-image. The set of all images (which is just a subset of the codomain) is called the range.

If we look at it, a function is just like a relation. It pairs two elements from two different sets. However, not all relations are functions. In fact, there are two specific properties that a function must satisfy:

1. every element in the domain must have an image (that is, every element in the domain is a pre-image);

2. every pre-image has exactly one image.

Trained students in mathematics would just simplify both conditions into one: every element in the domain has exactly one image.

In this lecture, we will restrict our coverage only to algebraic functions. The following is an example of a linear function which is one of the most basic types of algebraic functions. 

So when x = 1, 

So the input is 1, the output is 3, and the process (or relationship between the input and the output) is 2x + 1. 

Exercise. Let f(x) = x² - 2x + 1. Evaluate the function for the following values of x. 

Just like numbers, we can also operate functions. We can add, subtract, multiply, and divide them. There is also another operation on functions called composition which we will introduce later. 

If f(x) and g(x) are two functions, then 

For example, 

So if we are to find (f + g)(3), we have 

If f(x) and g(x) are two functions, then 

For example, 

So if we are to find (f - g)(-1), we have 

If f(x) and g(x) are two functions, then 

For example, 

So if we are to find (f g)(-1), we have 

If f(x) and g(x) are two functions, then 

For example, 

So if we are to find (f /g)(2), we have 

The last operation on functions we will be learning is composition denoted by ○. When we have (f ○ g) we read it as "the composite function f of g". It is defined by 

Hence, we can also read the composite function as "f of g of x". Here, g(x) is called the inner function while f(x) is called the outer function. By this definition, in order to evaluate (f ○ g)(x) we simply have to plugin g(x) into the "x" in f(x). For example, if 

Numbers can be classified into sets of numbers according to their properties. The table below lists the names, properties of and symbols used for the main number types.

Note: Many numbers are included in more than one set.

Recall the definition of a function: 

A function maps the set of pre-images, A (called the domain) into a set where the images, B are found (called the codomain).

Mathemtically, f : A → B

A binary operation, * is a function that maps the set of pre-images, AxA or A² (called the domain) into a set where the images, A are found (called the codomain).

Mathematically, * : A² → A

Regardless of what the answer to the problem is, we should avoid writing expressions in this way. As emphasized in Lecture 04 at the beginning of this chapter, mathematical language should be precise. It is supposedly clear and not vague. If a mathematical expression causes confusion, then it better be rewritten in a way that is better understood. 

Exercises:

1. Let ‘*’ be a binary operation on N defined by *(a,b) = a - b + a b2, then find 4 * 5. 

2. Let ‘*’ be a binary operation defined by a*b=4ab. Find (a*b)*a. 

3. Let ‘*’ be a binary operation defined by a*b=3ab+5. Find 8*3. 

4. Let '*' be a binary operation defined by m * n = (m/n - n/m). Find (-3 * 4)

5. Let ^ and * be binary operations defined by x ^ y = 2x + 2y + 8 and x * y = 3x2 + 4y3 - 2. Find (2*3) ^ (1*2)

You try this!

1. Let ‘*’ and "^" be binary operations defined as:

 *(x,y) = x2 - 2xy + y2 ; ^(x,y) = 2x + 3y - 4

Find (3*2) ^ (1*4). 

2. Let "#" and "%" be binary operations defined as:

#(x,y) = 3x + 2y + 6; %(x,y) = 5x2 - y2

Find (2%3) # (1%4)

3. Let "<" and ">" be binary operations defined as:

<(x,y) =  2x + y - 5; >(x,y) = x3 + y3 - 3xy

Find (2>1) < (4>3)

4. Let "&" and "@" be binary operations defined as: 

(x&y) = 4x + 5y - 7; (x@y) = x2 + y2 + x2y2

Find (1&3) @ (4&2)

5. Let "~" and "^" be binary operations defined as:

~(x,y) = x2 + y2 -  2x2y2; ^(x,y) = 7x2 + 8y2 + 3

Find: (3~2) ^ (4~1)

Read more on Closure Property here.

Related topics: Rational Numbers; Real Numbers; Integers; Natural Numbers

References

[1] Fraleigh, J. B. (2003). A first course in abstract algebra. Pearson Education India.

[2] Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (Vol. 3). Hoboken: Wiley.

[3] Nicholson, W. K. (2012). Introduction to abstract algebra. John Wiley & Sons.