NIOS 11th & 12th Math Odyssey

Chapter 1: Fundamentals of Mathematics & Integrated Functions

A single sign can change an answer completely: 3 - 5 is negative, while 3 - (-5) is positive. Mathematics becomes reliable when every symbol has a clear meaning and every operation follows a definite rule. The same discipline carries us from simple numbers to functions, graphs, inequalities, and eventually the advanced problems where several ideas work together.

The purpose here is to build that reliable base. We begin with the number system and algebraic language, then connect sets, relations, and functions. Finally, we combine these ideas with Class 10 algebra and selected 11th - 12th concepts so that mathematics is understood as one connected structure rather than a collection of separate chapters.

Real Numbers, Intervals, and Algebraic Language

The real numbers include every number that can be placed on a number line. Natural numbers are 1, 2, 3, and so on. Whole numbers include 0. Integers include negative numbers, zero, and positive numbers. Rational numbers can be written as p/q, where p and q are integers and q is not zero. Numbers such as √2 and π are irrational because they cannot be expressed in that form.

Together, rational and irrational numbers form the real numbers. This classification is useful because different operations behave differently in each group. For example, √2 is real, but √(-2) is not a real number; it belongs to the complex-number system studied later.

An interval describes a portion of the number line. The interval [2, 5] includes both endpoints, so 2 ≤ x ≤ 5. The interval (2, 5) excludes them, so 2 3 is written as (3, infinity), with 3 excluded.

Absolute value measures distance from zero. Therefore, |5| = 5 and |-5| = 5. The equation |x| = 4 has two solutions, x = 4 and x = -4, because both numbers are four units from zero. This idea becomes important when solving inequalities.

Consider |x - 2| 3, would the answer lie inside or outside this interval? It would lie outside: x 5.

Algebra uses letters to represent numbers. In the expression 4x^2 - 7x + 3, x is the variable, 4 is the coefficient of x^2, and 3 is the constant term. The laws of indices simplify powers:

a^m × a^n = a^(m+n)

a^m / a^n = a^(m-n), when a is not zero

(a^m)^n = a^(mn)

These laws require the same base. For example, 2^3 × 2^4 = 2^7, but 2^3 × 3^4 cannot be combined in this way.

Factorisation reverses multiplication. The expression x^2 - 5x + 6 becomes (x - 2)(x - 3), because the two numbers multiply to 6 and add to -5. This connects directly with quadratic equations. If (x - 2)(x - 3) = 0, then x = 2 or x = 3, using the zero-product rule.

Practical takeaway: Before manipulating an expression, identify its number type, interval meaning, and algebraic structure. A correct solution often begins with this classification.

Sets, Relations, and Mappings

A set is a well-defined collection of objects. The set A = {1, 2, 3, 4} contains four elements. Repeated elements are written only once, so {1, 2, 2, 3} is the same set as {1, 2, 3}. The symbol ∈ means “belongs to”; 2 ∈ A is true, while 5 ∈ A is false.

The union A ∪ B contains elements in A or B or both. The intersection A ∩ B contains elements common to both. The difference A - B contains elements in A but not in B. If the universal set is U, the complement of A, written A', contains all elements of U that are not in A.

For example, let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, and B = {3, 4, 5}. Then:

A ∪ B = {1, 2, 3, 4, 5}

A ∩ B = {3}

A - B = {1, 2}

A' = {4, 5, 6}

A relation connects elements of one set with elements of another. If A = {1, 2, 3} and B = {2, 4, 6}, the relation “is half of” connects 1 with 2, 2 with 4, and 3 with 6. Relations may be represented by ordered pairs, arrow diagrams, or a rule.

A function is a special relation in which every input has exactly one output. The input set is the domain, the allowed output values form the codomain, and the actual outputs form the range. For f(x) = x^2, if the domain is {-2, -1, 0, 1, 2}, the range is {0, 1, 4}. Notice that -2 and 2 have the same output; this is allowed. What is not allowed is one input having two different outputs.

The vertical-line test gives a graph-based check for a function. If any vertical line cuts a graph at more than one point, the graph does not represent y as a function of x.

A function may be combined with another function. If f(x) = 2x + 1 and g(x) = x^2, then:

(f + g)(x) = x^2 + 2x + 1

(fg)(x) = x^2(2x + 1)

(f/g)(x) = (2x + 1)/x^2, with x not equal to 0

The composite function f(g(x)) means that g acts first and f acts on the result. For the same functions:

f(g(x)) = f(x^2) = 2x^2 + 1

But g(f(x)) = (2x + 1)^2, which is generally different. Function composition is not usually commutative.

A function is one-one if different inputs produce different outputs. It is onto if every codomain element is reached....