Relations and Functions
Class 11 Mathematics · Chapter 2
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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: