What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Define an equivalence relation.

A relation \(R\) on a set \(A\) is an equivalence relation if it is reflexive (\(aRa\)), symmetric (\(aRb\Rightarrow bRa\)) and transitive (\(aRb,\,bRc\Rightarrow aRc\)).

When is a function one-one?

A function \(f\) is one-one if \(f(x_1)=f(x_2)\) implies \(x_1=x_2\); equivalently, distinct inputs map to distinct outputs.

When is a function onto?

A function \(f:A\to B\) is onto if for every \(y\in B\) there exists \(x\in A\) with \(f(x)=y\); the range equals the codomain.

Define a bijective function.

A function that is simultaneously one-one and onto; only bijections are invertible.

What is the composition \(g\circ f\)?

For \(f:A\to B\) and \(g:B\to C\), \((g\circ f)(x)=g(f(x))\), a function from \(A\) to \(C\).

State the condition for \(f^{-1}\) to exist.

\(f\) must be a bijection; then \(f^{-1}\) satisfies \(f^{-1}\circ f=I_A\) and \(f\circ f^{-1}=I_B\).

Is \(f(x)=x^2\) on \(\mathbb{R}\) one-one?

No, because \(f(2)=f(-2)=4\); distinct inputs share an output, so it is not one-one.

Show \(R=\{(a,b):a=b\}\) is an equivalence relation.

It is reflexive (\(a=a\)), symmetric (\(a=b\Rightarrow b=a\)) and transitive (\(a=b,b=c\Rightarrow a=c\)); hence an equivalence relation.

If \(f\) and \(g\) are one-one, is \(g\circ f\) one-one?

Yes; the composition of two one-one functions is one-one, and likewise composition of onto functions is onto.

What is an equivalence class?

For an equivalence relation \(R\) on \(A\), the equivalence class of \(a\) is \([a]=\{x\in A: xRa\}\); these classes partition \(A\).

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Chapter 1: Relations and Functions — Class 12 Mathematics

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Chapter 1: Relations and Functions

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A relation that is reflexive, symmetric and transitive is called:

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 \(f^{-1}\circ f=I\).

Important questions

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Equivalence relation

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A relation that is reflexive, symmetric and transitive.

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