Relations and Functions
Class 12 Mathematics · Mathematics Part I · Chapter 1
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Physics
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Physics
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اردو (کور)
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Chemistry
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Fine Art · Painting
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Informatics Practices
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Accountancy
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सृजनात्मक लेखन और अनुवाद
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हिंदुस्तानी संगीत (तबला / पखावज)
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Economics
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Economics
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History
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History
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History
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Geography
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Geography
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Geography
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English Elective · Kaleidoscope
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Political Science
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Political Science
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Sociology
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Sociology
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Psychology
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English Core · Flamingo
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English Core · Vistas
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Home Science
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Home Science
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Summary
This chapter deepens the study of relations and functions begun in Class 11. A relation on a set is examined for the properties of being reflexive, symmetric and transitive; a relation that is all three is called an equivalence relation, and it partitions the set into disjoint equivalence classes. The chapter then classifies functions as one-one (injective), onto (surjective) and bijective. A function \(f:A\to B\) is one-one if \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\), and onto if every element of \(B\) is the image of some element of \(A\). Composition of functions \((g\circ f)(x)=g(f(x))\) is introduced, and a bijection is shown to have an inverse \(f^{-1}\) satisfying \(f^{-1}\circ f=I\). These ideas form the backbone for inverse trigonometric functions and for understanding invertibility throughout calculus and algebra.
Types of relationsEquivalence relations and classesOne-one, onto and bijective functionsComposition of functionsInvertible functions
Key terms
Equivalence relation
A relation that is reflexive, symmetric and transitive.
One-one (injective)
Distinct inputs give distinct outputs: \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\).
Onto (surjective)
Every element of the codomain is an image of some element of the domain.
Bijective
A function that is both one-one and onto.
Composition
\((g\circ f)(x)=g(f(x))\), applying \(f\) then \(g\).
Invertible function
A bijection \(f\) for which \(f^{-1}\) exists with \(f^{-1}\circ f=I\).
Extra questions & answers
Try one from this chapter
A relation that is reflexive, symmetric and transitive is called: