The Four Basic Concepts of Mathematics

1. Sets

A set is a collection or grouping of well-defined and distinct objects, called elements or members, with no specific order. These elements can be anything, such as numbers, letters, or even other sets. The defining characteristic of a set is that each element appears only once within the collection. Sets are designated with capital letters, like A, B, or C.

To represent sets, we use braces { } to enclose the elements that belong to them. For example, if we have a set A that contains the numbers 1, 2, and 3, we would write it as A = {1, 2, 3}.

The symbol used to indicate that an object is an element of a set is “∈”. When we say “x ∈ A”, it means that the element x is a member of the set A. This notation signifies the relationship between an element and a set and is commonly used in mathematics to express membership. Conversely, if an element is not a member of a set, we use the symbol “∉” or “not ∈”. For example, if we have a set A = {1, 2, 3}, we can say that 2 ∈ A, but 4 ∉ A.

a. Given Sets: A = {1, 2 ,3 ,4 ,5} B = {2, 4, 6, 8}

• Problem: Determine if 3 ∈ A.
• Answer: Yes, 3 is an element of set A.

b. Given Sets: X = { a, e, i, o, u}

• Problem: Determine if ‘e’ ∈ X.
• Answer: Yes, ‘e’ is an element of set X.

c. Given Sets: Y = {apple, banana, cherry}

• Problem: Determine if ‘grape’ ∈ Y.
• Answer: No, ‘grape’ is not an element of set Y.

d. Given Sets: Z = {2, 4, 6, 8, 10}

• Problem: Determine if 7 ∈ Z.
• Answer: No, 7 is not an element of set Z.

e. Given Sets: M = {red, blue, green, yellow}

• Problem: Determine if ‘green’ ∈ M.
• Answer: Yes, ‘green’ is an element of set M.

These problems demonstrate the basic usage of the ∈ symbol in determining membership within a set.

Sets provide a foundation for various mathematical operations and concepts, such as intersections, unions, subsets, and more. They help us organize and study collections of objects or elements within mathematics.

1.1. There are two ways to describe a set, namely:

In the Roster/Tabular Method, the elements of the specified set are presented within a pair of braces, separated by commas for enumeration.

Example: Let’s consider the set of primary colors.

• Roster Method representation: {red, blue, yellow}
• Elements of the set: {red, blue, yellow}

Note: In this example, we used words to represent the elements, but the elements could also be numbers, symbols, or other objects.

In the Rule/Descriptive Method, the elements of a set are defined based on a common characteristic. This approach employs set builder notation, where the variable “x” is used to represent any element within the given set.

Example: Let’s consider the set of even numbers less than 10.

• Rule Method representation: {x | x is an even number and x < 10}
• Elements of the set: {2, 4, 6, 8}

It is worth mentioning that both methods are different notations for representing sets, and the choice of method depends on the context and purpose of the set being described.

1.2. The following are kinds of sets:

1. Finite Set. A finite set is a set that contains a specific and countable number of elements. For example:

• Set A = {1, 2, 3, 4, 5} is a finite set with five elements.
• Set B = {apple, banana, cherry} is a finite set with three elements.

2. Infinite Set. An infinite set is a set that contains an uncountable number of elements. For example:

• Set of natural numbers, N = {1, 2, 3, …}, is an infinite set as it goes on indefinitely.
• Set of real numbers, R, is also an infinite set.

3. Empty Set. Also known as a null set, it is a set that contains no elements. It is denoted by the symbol “∅” or “{}”. For example:

• Set C = ∅ is an empty set.

4. Universal Set. Is a set that contains all the elements or objects under consideration in a particular context or problem. It is often denoted by the symbol “U”. The elements of the universal set may vary depending on the specific problem or situation. Here are a few examples to help illustrate the concept:

• In the context of students in a classroom, the universal set might be all the students present in that classroom. So, if there are 30 students in the class, the universal set would contain all 30 students.
• If you’re considering the set of all positive integers, the universal set would be the set of all numbers greater than zero (1, 2, 3, 4, 5, …).
• In the context of colors, the universal set might be all the colors in the visible spectrum (red, orange, yellow, green, blue, indigo, violet, etc.).

1.3. Two or more sets may be related to each other as described by the following:

1. Equal Sets. Two sets are considered equal if they have the same elements. In other words, if every element in set A is also present in set B, and vice versa, then A and B are equal sets. The order of the elements does not matter. For example:

• Set A = {1, 2, 3} and set B = {2, 1, 3} are equal sets because they have the same elements, even though the order is different.

2. Equivalent Sets. Two sets are considered equivalent if they have the same number of elements. In other words, if the cardinality (number of elements) of set A is equal to the cardinality of set B, then A and B are equivalent sets. The actual elements in the sets may differ. For example:

• Set C = {apple, orange, banana} and set D = {dog, cat, bird} are equivalent sets, as both have three elements.

3. Joint Sets. Joint sets refer to the collection of all elements that are common to two or more sets. For example:

• Set E = {1, 2, 3} and set F = {2, 3, 4} have a joint set, which is {2, 3}. These elements are present in both sets.
• The symbol used to represent joint sets is the intersection symbol (∩). 

4. Disjoint Sets. Disjoint sets, on the other hand, are sets that have no elements in common. In other words, if the intersection of two sets is an empty set, then they are disjoint sets. For example:

• Set G = {1, 2, 3} and set H = {4, 5, 6} are disjoint sets because they do not share any common elements.
• The symbol used to represent disjoint sets is the null set symbol (∅) or sometimes the symbol for “does not intersect” (⊄).

It’s important to note that joint sets and disjoint sets are mutually exclusive. If two sets are joint, they cannot be disjoint, and vice versa.

It can also be noted that equal sets are equivalent sets, however, not all equivalent sets are equal sets.

A subset refers to a set in which every element can be found within a larger set. The symbol “⊆” denotes “subset,” while “⊈” signifies “not a subset of.”

1. Proper Subset. A proper subset is a subset that contains some, but not all, of the elements of another set. In other words, every element of the proper subset is also an element of the larger set, but the larger set contains at least one additional element.

For example, let’s consider two sets: A = {1, 2, 3, 4} B = {1, 2}

In this case, B is a proper subset of A because every element in B (1 and 2) is also an element of A, but A contains additional elements (3 and 4) that are not in B.

2. Improper Subset. An improper subset is a subset that contains all the elements of another set. In other words, every element of the improper subset is also an element of the larger set, and there are no additional elements in the larger set.

For example, let’s consider the same sets as before: A = {1, 2, 3, 4} B = {1, 2, 3, 4}

In this case, B is an improper subset (or simply the same set) of A because every element in B is also an element of A, and there are no additional elements in A.

A null set always qualifies as a subset of any given set and is considered an improper subset. With the exception of the set itself and the null set, all other subsets are considered proper subsets. The power set, which contains all subsets of a given set with “n” elements, comprises 2ⁿ elements.

1.4. Four set operations are commonly performed when working with sets. Let’s consider two sets named Set A and Set B.

1. Union. The union of two sets combines all the elements from both sets, removing any duplicates. It is denoted by the symbol (∪).

• For example: Set A = {1, 2, 3} Set B = {3, 4, 5}
• The union of A and B is A ∪ B = {1, 2, 3, 4, 5}

2. Intersection. The intersection of two sets includes only the elements that are common to both sets. It is denoted by the symbol (∩).

• For example: Set C = {1, 2, 3} Set D = {3, 4, 5}
• The intersection of C and D is C ∩ D = {3}

3. Difference. The difference of two sets gives us the elements that are present in one set but not in the other. It is denoted by the symbol (−) or by using the word “except”.

• For example: Set E = {1, 2, 3, 4, 5} Set F = {3, 4}

4. Complement. The complement of a set represents the elements that are not present in that set but are part of a universal set. It is often used in the context of a universal set.

• For example: Universal set U = {1, 2, 3, 4, 5} Set G = {3, 4}
• The complement of G with respect to U is G’ = {1, 2, 5}
• The difference of E and F is E – F = {1, 2, 5}

These four set operations provide valuable tools for comparing, combining, and manipulating sets, allowing us to analyze their relationships and properties. I hope these examples help clarify the different set operations. Let me know if you have any further questions!

Venn-Euler Diagrams, commonly known as Venn Diagrams, are used to visually depict the relationships and operations of sets. Typically, a rectangle is used to represent the universal set, while circles within the rectangle represent its subsets. The shaded region in these diagrams represents the relationship or operation between the sets.

2. Relations

A relation in mathematics is a set of ordered pairs that express a relationship between two sets of objects or elements. It defines how the elements from one set are related to the elements of another set.

For example, suppose we have two sets: A = {1, 2, 3} and B = {a, b, c}. We can establish a relation between them by pairing the elements together, such as {(1, a), (2, b), (3, c)}. This relation indicates that element 1 in set A is related to element ‘a’ in set B, element 2 in set A is related to element ‘b’ in set B, and so on.

Relations can also be depicted graphically as arrows or lines connecting the elements from one set to another. They can represent different types of relationships, such as one-to-one, many-to-one, one-to-many, or many-to-many.

See the figures:

a. Disjoint set A and B

b. Set A is a proper subset of Set B, A ⊆ B

c. Union of sets A and B, A ꓴ B

d. Intersection of sets A and B, A ꓵ B

e. Difference of sets A and B, A – B

f. Complement of a set A, A’

A relation can be visually represented through a mapping diagram or a graph. These methods illustrate the correspondence from the domain to the range, showcasing the pairing in the relation.

For instance, an example of representing a relation is by:

in which the lines connect the inputs with their outputs. The relation can also be represented as:

It is important to note that a relation does not necessarily have to follow a specific pattern or rule like a function does. Each element in the domain of a relation may or may not be associated with a unique element in the range. However, in the case where each element in the domain is associated with one and only one element in the range, the relation is classified as a function.

3. Functions

In mathematics, a function is a rule or relationship between two sets of numbers, known as the domain and the range. It assigns each element in the domain to exactly one element in the range.

In the given example, the function ‘f’ establishes a mapping between the four students (Alyssa, Elijah, Steph, and Shei) in set X and their respective favorite subjects (Chemistry, Math, Physics, and Statistics) in set Y. This means that each student from set X is uniquely paired with their corresponding favorite subject from set Y.

X = {Alyssa, Elijah, Steph, Shei}

Y = {Chemistry, Math, Physics, Statistics}

We represent this pairing using ordered pairs, denoted as (x, y), where x represents an element from set X and y represents the corresponding element from set Y. The set of ordered pairs in this example is {(Alyssa, Chemistry), (Elijah, Math), (Steph, Physics), (Shei, Statistics)}.

Functions are mathematical entities that give unique outputs to particular inputs.

There are many ways to think about functions, but they, at all times, have three most important parts:

1. Input
2. Relationship
3. Output

Additionally, there are specific rules that apply to functions.

• Functions handle all input values.
• Functions maintain a single relationship per input.

Furthermore, a function comprises various components, including the argument (input provided to the function), value (output obtained from the function), domain (the set of permissible input values for a given function), and codomain (the set of allowable output values).

To illustrate this concept, let’s consider the set X, which consists of four shapes: triangle, rectangle, hexagon, and square. Similarly, let’s take the set y, which comprises five colors: red, blue, green, pink, and yellow.

The “color-of-the-shape function” establishes a connection between each shape and its respective color, serving as a mapping from X to Y. Each shape is associated with a unique color, ensuring that no shape is without a color and no shape has more than one color. The function’s domain consists of the four shapes, while its codomain encompasses the five available colors. It is important to note that for a function, not every possible output needs to correspond to a specific input. For example, the color blue does not represent any of the four shapes in X (Function, 2017).

There are several rules to keep in mind when performing operations on functions. These rules guide us in manipulating functions and simplifying expressions. Here are some common rules:

1. Addition and Subtraction of Functions

When adding or subtracting two functions, you combine like terms. For example: (f + g)(x) = f(x) + g(x).

Similarly, when subtracting two functions, you subtract corresponding terms. For example: (f – g)(x) = f(x) – g(x).

2. Multiplication of Functions

When multiplying two functions, you can apply the distributive property. For example: (f * g)(x) = f(x) * g(x).

3. Division of Functions

Division of functions is performed similarly to multiplication but with the reciprocal of the divisor. For example: (f / g)(x) = f(x) / g(x).

Composition of Functions

The composition of two functions, denoted as (f ∘ g)(x), involves substituting one function into another. The output of one function becomes the input of the other. For example: (f ∘ g)(x) = f(g(x)).

Domain Restrictions

When performing operations on functions, it’s important to consider the domain restrictions. For instance, division by zero or evaluating a function where it’s undefined should be avoided.

These are general rules, but the specific operations and rules may vary depending on the nature of the functions and the type of operation being performed. It’s crucial to always follow the specific rules associated with the given functions or operation.

4. Binary Operations

Binary operations are fundamental mathematical operations that involve two operands and produce a single result. These operations are widely used in various mathematical fields, computer science, and everyday life. There are several common binary operations, and each has its own properties and applications. Let’s discuss them in detail:

A. Addition (+)

• Definition. Addition is the process of combining two numbers to produce their sum.
• Notation. a + b, where ‘a’ and ‘b’ are the operands.
• Properties
  • Commutative. a + b = b + a
  • Associative. (a + b) + c = a + (b + c)
  • Identity Element. 0 (a + 0 = a)
  • Inverse Element. -a (a + (-a) = 0)
• Applications. Used for counting, measuring, and arithmetic calculations.

B. Subtraction (-)

• Definition. Subtraction is the process of taking one number away from another to produce the difference.
• Notation. a – b, where ‘a’ is the minuend, and ‘b’ is the subtrahend.
• Properties
  • Not commutative. a – b ≠ b – a
  • Not associative. (a – b) – c ≠ a – (b – c)
• Applications. Used for finding the difference between two quantities.

B. Multiplication (×)

• Definition. Multiplication is the process of repeatedly adding one number (the multiplicand) to itself a certain number of times (the multiplier).
• Notation. a × b or ab, where ‘a’ and ‘b’ are the operands.
• Properties.
  • Commutative. a × b = b × a
  • Associative. (a × b) × c = a × (b × c)
  • Identity Element. 1 (a × 1 = a)
  • Distributive over addition. a × (b + c) = (a × b) + (a × c)
• Applications. Used for calculating areas, volumes, and various mathematical operations.

C. Division (÷ or /)

• Definition. Division is the process of splitting a quantity into equal parts or finding out how many times one number is contained within another.
• Notation. a ÷ b or a / b, where ‘a’ is the dividend, and ‘b’ is the divisor.
• Properties
  • Not commutative. a ÷ b ≠ b ÷ a
  • Not associative. (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
• Applications. Used for calculating rates, ratios, proportions, and solving equations.

D. Exponentiation (^ or )

• Definition. Exponentiation is the operation of raising a base number to a certain power.
• Notation. a^b or a**b, where ‘a’ is the base, and ‘b’ is the exponent.
• Properties.
  • Not commutative: a^b ≠ b^a
  • Not associative: (a^b)^c ≠ a^(b^c)
  • Identity Element: 1 (a^0 = 1)
• Applications. Used for expressing repeated multiplication, compound interest, and in various mathematical and scientific formulas.

E. Modulo (%) or Remainder (mod).

• Definition. Modulo is the operation that returns the remainder when one number is divided by another.
• Notation. a % b or a mod b, where ‘a’ is the dividend, and ‘b’ is the divisor.
• Properties
  • Not commutative: a % b ≠ b % a
  • Not associative: (a % b) % c ≠ a % (b % c)
• Applications. Used for determining divisibility, finding cyclic patterns, and in cryptography.

Binary System

The binary system, often referred to as the binary numeral system or base-2 numeral system, is a way of representing numbers using only two symbols: 0 and 1. The binary system is fundamental in the realm of computers and digital systems, as it’s the underlying representation of data in these systems.

Here’s a detailed discussion of the binary system:

1. Basics

• In the decimal (base-10) system, which is the most familiar numeral system, numbers are represented using ten symbols: 0, 1, 2, …, 9. Each position in a decimal number represents a power of 10, with the rightmost position representing 100100, the next one 101101, and so on.
• In contrast, in the binary system, there are only two symbols: 0 and 1. Each position in a binary number represents a power of 2. The rightmost position represents 2020, the next one 2121, and so forth.

2. Representation. To better understand, let’s take the binary number 1101. Its decimal equivalent can be found as:

(1×2³) + (1×2²) + (0×2¹) + (1×2⁰)

8 + 4 + 0 + 1 = 13

So, the binary number 1101 represents the decimal number 13.

3. Conversion from Decimal to Binary. Converting a decimal number to binary involves repeatedly dividing the number by 2 and keeping track of the remainders:

• For the number 13:
  1. 13 divided by 2 gives quotient 6, remainder 1.
  2. 6 divided by 2 gives quotient 3, remainder 0.
  3. 3 divided by 2 gives quotient 1, remainder 1.
  4. 1 divided by 2 gives quotient 0, remainder 1.

Reading the remainders upwards gives the binary representation: 1101.

4. Arithmetic in Binary. Binary arithmetic is similar to decimal arithmetic but simpler because there are only two digits.

• Addition. 1101 (13 in decimal) + 1011 (11 in decimal) 1000 (Carry) 1000 (Result: 24 in decimal) +​1111​1000​0100​1100 ​(13 in decimal) (11 in decimal) (Carry) (Result: 24 in decimal)​​
• Subtraction, multiplication, and division can also be performed using binary digits, albeit with their unique methods compared to the decimal system.

5. Significance in Computing. The binary system is fundamental to modern computing for several reasons:

• Physical Implementation. Computer systems, at their most fundamental level, have electronic circuits that detect the presence or absence of electric current—two states. These two states map naturally to the binary system (1 for the presence of current and 0 for its absence).
• Logical Implementation. Binary logic, through gates like AND, OR, NOT, XOR, etc., forms the basis for computation and data manipulation in computers.
• Data Representation. All data, be it numbers, text, images, or anything else, is stored, processed, and transmitted as binary data in digital systems.

6. Other Forms. Binary representations have led to other related forms, such as:

• Binary-Coded Decimal (BCD). Where each decimal digit is represented by a fixed number of binary digits.
• Octal and Hexadecimal Systems. These are base-8 and base-16 systems, respectively, used in computing as shorthand notations for binary numbers since their bases are powers of 2.

In summary, the binary system is a numeral system with base-2. It’s essential in digital electronics and computing because it aligns perfectly with the on-off nature of electronic signals. Moreover, the concepts from binary arithmetic and logic form the cornerstone of all computational processes.

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Elementary Logic

Logic, often referred to as the science of formal principles of reasoning or correct inference, is a field that revolves around studying and understanding the principles and methods necessary to differentiate valid arguments from invalid ones. Its application is significant in various areas of human pursuits.

Mathematical logic focuses on studying how reasoning works within mathematics. In mathematics, reasoning is deductive, which means that you draw conclusions based on given hypotheses or statements. The main idea is that one statement can logically follow from others.

When we use the term “therefore,” it typically indicates that a statement is a consequence of the preceding statements. This helps to establish a logical flow in our reasoning process.

Logic is a way of expressing our thoughts in an organized manner, starting from basic assumptions (called axioms) and leading to a conclusion. In mathematical logic, there are specific rules and formalities that ensure the preservation of truth throughout the logical argument. These rules help us maintain consistency and accuracy when making deductions.

Once we have successfully built a conclusion using logical steps, we can confidently use that conclusion as an axiom in a different logical argument. This allows us to build upon our previous reasoning and expand our understanding.

In logic, an argument is defined as a group of statements or propositions, where one statement (the conclusion) is claimed to follow from the others (the premises). The premises are then used as the basis for asserting the truth of the conclusion.

In the context of categorical syllogism, which is a specific type of argument, there are two premises and a single conclusion. This form of argument allows us to establish relationships between categories or classes based on logical deductions.

Formality

Formality is a concept that exists in relation to other expressions, meaning one expression can be more or less formal compared to another, establishing an ordering of expressions. However, it is important to note that no expression can be completely formal or completely informal. All linguistic expressions fall somewhere on a spectrum between absolute formality and absolute informality.

A formal expression is considered fully formal when it possesses three key characteristics: context-independence, precision, and unambiguity. This means that the expression maintains a clear and consistent meaning that does not change regardless of the context in which it is used.

When we aim to test the validity of knowledge, it often becomes necessary to express it formally. This formality refers to the ability of an expression to retain its meaning under various contexts. Both mathematical formalism and operational determination contribute to the formality of an expression.

Formal expressions offer several advantages, including storability, universal communicability, and testability. Storability refers to the ability to store and retrieve these expressions accurately. Universal communicability signifies that these expressions can be understood and shared by anyone who is familiar with the formal system. Testability means that these expressions can be subjected to logical tests to ascertain their truth or consistency.

However, achieving complete formality is unattainable because context plays a significant role. In formal or operational definitions, certain aspects are inherently tied to a specific context. These aspects include primitive terms, observation setups, and background conditions. These elements exist outside the formal system and are necessary for understanding the meaning and application of formal expressions.

Therefore, formality is not an absolute property but rather dependent on the context in which the expressions are used. Different individuals may apply varying levels of formality in different situations or for various purposes, based on the specific context they are operating within.

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