What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

If \(n(A)=3\) and \(B=\{3,4,5\}\), how many elements has \(A\times B\)?

It has \(n(A)\times n(B)=3\times 3=9\) ordered pairs.

Define a function from set \(A\) to set \(B\).

A function \(f:A\to B\) is a relation in which every element of \(A\) has one and only one image in \(B\); no two ordered pairs have the same first element.

Give the number of relations from \(A\) to \(B\) when \(n(A)=p,\ n(B)=q\).

Since \(n(A\times B)=pq\), the number of relations (subsets of \(A\times B\)) is \(2^{pq}\).

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

For the root to be real, \(9-x^2\ge 0\Rightarrow -3\le x\le 3\); domain \([-3,3]\), range \([0,3]\).

Describe the signum function and its range.

The signum function gives \(1\) for \(x>0\), \(0\) for \(x=0\) and \(-1\) for \(x<0\); its domain is \(\mathbb{R}\) and range is \(\{-1,0,1\}\).

Write the greatest integer function and the values of \([2.7]\) and \([-1.2]\).

\(f(x)=[x]\) is the greatest integer not exceeding \(x\); \([2.7]=2\) and \([-1.2]=-2\).

If \((x+1,\,y-2)=(3,1)\), find \(x\) and \(y\).

Equating components, \(x+1=3\) and \(y-2=1\), so \(x=2\) and \(y=3\).

Is \(A\times B=B\times A\) in general? Explain.

No. \((a,r)\ne(r,a)\) as ordered pairs differ, so in general \(A\times B\ne B\times A\), though both have the same number of elements.

For \(f(x)=x^2\) and \(g(x)=2x+1\), write \((fg)(x)\).

By pointwise multiplication, \((fg)(x)=x^2(2x+1)=2x^3+x^2\).

Define a real function and a real valued function.

A real valued function has \(\mathbb{R}\) or a subset of \(\mathbb{R}\) as range; if its domain is also \(\mathbb{R}\) or a subset of \(\mathbb{R}\), it is a real function.

Relations and Functions

Q: Write the greatest integer function and the values of \([2.7]\) and \([-1.2]\).

\(f(x)=[x]\) is the greatest integer not exceeding \(x\); \([2.7]=2\) and \([-1.2]=-2\).

Q: If \((x+1,\,y-2)=(3,1)\), find \(x\) and \(y\).

Equating components, \(x+1=3\) and \(y-2=1\), so \(x=2\) and \(y=3\).

Q: Is \(A\times B=B\times A\) in general? Explain.

No. \((a,r)\ne(r,a)\) as ordered pairs differ, so in general \(A\times B\ne B\times A\), though both have the same number of elements.

Q: For \(f(x)=x^2\) and \(g(x)=2x+1\), write \((fg)(x)\).

By pointwise multiplication, \((fg)(x)=x^2(2x+1)=2x^3+x^2\).

Q: Define a real function and a real valued function.

A real valued function has \(\mathbb{R}\) or a subset of \(\mathbb{R}\) as range; if its domain is also \(\mathbb{R}\) or a subset of \(\mathbb{R}\), it is a real function.

Class 11 Mathematics · Chapter 2

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

View all 44 amazing Marks Booster kits →

Summary

This chapter links pairs of objects from two sets. The Cartesian product \(A\times B=\{(a,b):a\in A,\ b\in B\}\) collects all ordered pairs; two ordered pairs are equal iff their first and second elements match, and if \(n(A)=p,\ n(B)=q\) then \(n(A\times B)=pq\). A relation \(R\) from \(A\) to \(B\) is any subset of \(A\times B\); its domain is the set of first elements, its range the set of second elements, with the whole \(B\) as codomain (range \(\subset\) codomain). The total number of relations is \(2^{pq}\). A function is a special relation in which every element of the domain has exactly one image. The book studies real functions and graphs of the identity, constant, polynomial, rational, modulus \(|x|\), signum and greatest-integer \([x]\) functions. It also defines algebra of real functions: for \(f,g:X\to\mathbb{R}\), \((f+g)(x)=f(x)+g(x)\), \((f-g)(x)=f(x)-g(x)\), \((fg)(x)=f(x)g(x)\), \((kf)(x)=kf(x)\) and \((f/g)(x)=f(x)/g(x)\) when \(g(x)\ne 0\). These ideas make functions a precise language for the rest of mathematics.

Cartesian product of setsOrdered pairs and equalityRelations: domain, range, codomainFunctions as special relationsIdentity, constant and polynomial functionsModulus, signum and greatest integer functionsAlgebra of real functions

Key terms

Ordered pair

A pair \((a,b)\) in which order matters; \((a,b)=(c,d)\) iff \(a=c,\ b=d\).

Cartesian product

\(A\times B=\{(a,b):a\in A,\ b\in B\}\), with \(n(A\times B)=n(A)\,n(B)\).

Relation

A subset of \(A\times B\) describing a link between elements of \(A\) and \(B\).

Function

A relation in which each element of the domain has one and only one image.

Domain, range, codomain

Set of first elements (domain), set of second elements (range), and the whole set \(B\) (codomain).

Modulus function

\(f(x)=|x|\), equal to \(x\) for \(x\ge 0\) and \(-x\) for \(x<0\).

Extra questions & answers

Try one from this chapter

If \(n(A)=3\) and \(n(B)=2\), then \(n(A\times B)\) is:

Flashcards & interactive practice