Notes

What are real numbers?
- The real numbers in mathematics are the set of all numbers that possess no imaginary parts. They can also be defined as the set of numbers that arise from joining the set of rational numbers and the set of irrational numbers.
- In mathematics, real numbers are defined as the combination of rational and irrational numbers.

Search something or about real numbers.
- Rational numbers - are any numbers that can be represented by a fraction: a/b where both a,b are integers and b ≠ 0.
ex:
• integers (-2, 0, 1)
• fractions(1/2, 2/5)

- Irrational numbers - are simply not rational numbers in that they cannot be represented as a fraction of two integers.
ex:
• Numbers like pi and e are irrational.
• √3, π(22/7), etc., are all real numbers.

Imaginary Numbers - are numbers that, when squared, give a negative result. The most basic imaginary number is denoted by the symbol 'i', which is defined as the square root of -1.

Define relation and function
Relation - A relation is a relationship between sets of values. In math, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.

Function - A function in maths is a special relationship among the inputs (i.e. the domain) and their outputs (known as the codomain) where each input has exactly one output, and the output can be traced back to its input.

What are the types of functions?
1. Injective (One-to-One) Functions: A function in which one element of the Domain Set is connected to one element of the Co-Domain Set.

2. Surjective (Onto) Functions: A function in which every element of Co-Domain Set has one pre-image.

Example: Consider, A = {1, 2, 3, 4}, B = {a, b, c} and f = {(1, b), (2, a), (3, c), (4, c)}.

Therefore, It is a Surjective Function, as every element of B is the image of some A.

3. Bijective (One-to-One Onto) Functions: A function which is both injective (one to - one) and surjective (onto) is called bijective (One-to-One Onto) Function.

Example:
Consider P = {x, y, z}
Q = {a, b, c}
and f: P → Q such that
f = {(x, a), (y, b), (z, c)}

Therefore, The f is a one-to-one function and also it is onto. So it is a bijective function.

4. Into Functions: A function in which there must be an element of co-domain Y does not have a pre-image in domain X.

Example:
Consider, A = {a, b, c}
B = {1, 2, 3, 4} and f: A → B such that
f = {(a, 1), (b, 2), (c, 3)}
In the function f, the range i.e., {1, 2, 3} ≠ co-domain of Y i.e., {1, 2, 3, 4}

Therefore, it is an into function.

5. One-One Into Functions: Let f: X → Y. The function f is called one-one into function if different elements of X have different unique images of Y.

Example:
Consider, X = {k, l, m}
Y = {1, 2, 3, 4} and f: X → Y such that
f = {(k, 1), (l, 3), (m, 4)}

Therefore, The function f is a one-one into function.

6. Many-One Functions: Let f: X → Y. The function f is said to be many-one functions if there exist two or more than two different elements in X having the same image in Y.

Example:
Consider X = {1, 2, 3, 4, 5}
Y = {x, y, z} and f: X → Y such that
f = {(1, x), (2, x), (3, x), (4, y), (5, z)}

Therefore, The function f is a many-one function.

7. Many-One Into Functions: Let f: X → Y. The function f is called the many-one function if and only if is both many one and into function.

Example:
Consider X = {a, b, c}
Y = {1, 2} and f: X → Y such that
f = {(a, 1), (b, 1), (c, 1)}

Therefore, As the function f is a many-one and into, so it is a many-one into function.

8. Many-One Onto Functions: Let f: X → Y. The function f is called many-one onto function if and only if is both many one and onto.

Example:
Consider X = {1, 2, 3, 4}
Y = {k, l} and f: X → Y such that
f = {(1, k), (2, k), (3, l), (4, l)}

Therefore, The function f is a many-one (as the two elements have the same image in Y) and it is onto (as every element of Y is the image of some element X). So, it is many-one onto function.

What are the types of relations?
1. Empty Relation
A relation R on a set A is called Empty if the set A is an empty set, i.e. any relation where no element of set A is not related to the element of set B then it is called an empty relation.

For example, A = {1, 2, 3} and B = {5, 6, 7} where, R = {(x, y) where x + y = 22}, then it is an empty relation.

2. Reflexive Relation
A relation R on a set A is called reflexive if (a, a) ∈ R holds for every element a∈ A . i.e. if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

For example, A = {2, 3} then the reflexive relation R on A is,
• R = {(2, 2), (3, 3)}

3. Symmetric Relation
A relation R on a set A is called symmetric if (b, a) ∈ R holds when (a, b) ∈ R i.e. The relation R = {(a, b), (b, a)} is a reflexive relation on (a, b)

For example, A = {2, 3} then symmetric relation R on A is,
• R = {(2, 3), (3, 2)}

4. Transitive Relation
A relation R on a set A is called transitive if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R for all a,b,c ∈ A i.e.

For example, set A = {1, 2, 3} then transitive relation R on A is,
• R = {(1, 2), (2, 3), (1, 3)}

5. Equivalence Relation
A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive. i.e. relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)} on set A = {1, 2, 3} is equivalence relation as it is reflexive, symmetric, and transitive.

6. Universal Relation
Universal relation is a relation in which all elements of set are mapped to another element of set then it is called universal relation.

For example, A = {4, 8, 12} and B = {1, 2, 3} then the universal relation is, R = {(x, y) where x > y}.

7. Identity Relation
Identity relation is a relation defined such as all elements in a set are related to itself. It is defined as, I = {(x, x) : for all x ∈ X}.

For example P = {1, 2, 3} then Identity Relation(I) = {(1, 1), (2, 2), (3, 3)}.

8. Inverse Relation
A relation is called the inverse of any relation if elements of one set are inverse pairs of another set. The inverse of a relation R is denoted as R-1. i.e., R-1 = {(y, x): (x, y) ∈ R}.