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अध्याय 2: संबंध एवं फलन (Relations & Functions)

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CBSE • कक्षा XI • Mathematics • अध्याय 2

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

संबंध एवं फलन (Relations & Functions)

In Class 11 Mathematics, "Relations and Functions" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

🔄 Have You Ever Wondered?

What is the exact mathematical definition of a 'black box' machine that takes an input number, processes it, and spits out a unique output without fai...

What is the exact mathematical definition of a 'black box' machine that takes an input number, processes it, and spits out a unique output without fail? Functions are the operational gears of higher calculus.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Mathematics, "Relations and Functions" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Ordered pairs and Cartesian plane.
Set theory from Chapter 1.
Algebraic equations.

इस अध्याय के लक्ष्य

Define Cartesian Product of Sets ($A \times B$) and compute its cardinality ($|A \times B| = pq$).
Define a Relation as a subset of $A \times B$; identify Domain, Codomain, and Range.
Define a Function as a special relation where every element of domain has a unique image.
Analyze standard functions and graphs: Identity, Constant, Polynomial, Rational, Modulus ($|x|$), Signum, Greatest Integer ($[x]$).
Perform Algebra of Real Functions: $(f + g)(x)$, $(f - g)(x)$, $(fg)(x)$, and $(f/g)(x)$.

अध्याय रूपरेखा एवं प्रगति

1 1. Cartesian Product & Relations

2 2. Definition of a Function

3 3. Special Real Functions

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Cartesian Product & Relations

Given non-empty sets $A$ and $B$, the Cartesian Product is: $$\mathbf{A \times B = \{(a, b) : a \in A, b \in B\}} \quad (|A \times B| = |A| \cdot |B|)$$ A Relation $R$ from $A$ to $B$ is any subset of $A \times B$. The total number of possible relations is $\mathbf{2^{pq}}$ where $|A|=p, |B|=q$.

2. Definition of a Function

A relation $f$ from $A$ to $B$ is a Function if and only if:
• Every element of $A$ has an image in $B$, AND
• No element of $A$ has more than one image in $B$ (Vertical Line Test!).

3. Special Real Functions

Modulus Function: $f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$, Domain $=\mathbb{R}$, Range $=[0, \infty)$.
Signum Function: $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$, Domain $=\mathbb{R}$, Range $=\{-1, 0, 1\}$.
Greatest Integer Function: $f(x) = [x]$ (step function rounding down to largest integer $\le x$).

चित्रात्मक व्याख्या एवं मॉडल

Relations and Functions Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

1. Cartesian Product & Relations • 2. Definition of a Function

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Cartesian Product: Ordered pairs $(a, b)$ forming total relational coordinate space.

मुख्य बिंदु 2

Function: Mapping where every domain element is assigned a strictly unique range element.

मुख्य बिंदु 3

Vertical Line Test: Geometric check where any vertical line intersects a function graph at most once.

मुख्य बिंदु 4

Modulus: V-shaped graph outputting non-negative magnitude.

मुख्य बिंदु 5

Greatest Integer Function: Step-stair graph rounding down to floor integer.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

If $A = \{1, 2\}$ and $B = \{3, 4\}$, write $A \times B$ and state the total number of relations from $A$ to $B$.

उत्तर एवं व्याख्या देखें

उत्तर: $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$. Number of elements $= 2 \times 2 = 4$. Total relations $= 2^4 = 16$.
A × B has 4 pairs; 16 relations.

Find the domain and range of the real function $f(x) = \sqrt{9 - x^2}$.

उत्तर एवं व्याख्या देखें

उत्तर: For $f(x)$ to be real, $9 - x^2 \ge 0 \implies x^2 \le 9 \implies -3 \le x \le 3$. Domain $=[-3, 3]$. Since $0 \le \sqrt{9-x^2} \le 3$, Range $=[0, 3]$.
Domain = [-3, 3]; Range = [0, 3].

What is the Signum Function? State its domain and range.

उत्तर एवं व्याख्या देखें

उत्तर: The signum function is defined as $f(x) = \frac{|x|}{x}$ for $x \ne 0$ and $f(0) = 0$. Domain is $\mathbb{R}$; Range is $\{-1, 0, 1\}$.
Domain = R; Range = {-1, 0, 1}.

Evaluate $[3.7]$, $[-2.3]$, and $[5]$ for the greatest integer function $[x]$.

उत्तर एवं व्याख्या देखें

उत्तर: $[3.7] = 3$; $[-2.3] = -3$ (largest integer $\le -2.3$); $[5] = 5$.
[3.7]=3, [-2.3]=-3, [5]=5.

Why is every function a relation, but not every relation a function?

उत्तर एवं व्याख्या देखें

उत्तर: Because a function is a specialized relation with strict constraints: every element of the first set must relate to exactly one unique element in the second set; general relations permit one-to-many pairings.
Functions require unique single outputs for each input.

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