Relations and Functions

Mathematics is the art of giving the same name to different things. — Henri Poincaré

2.1 Ordered Pairs and Cartesian Product

An ordered pair $(a,b)$ has first component $a$ and second $b$. $(a,b)=(c,d)$ iff $a=c$ and $b=d$.

Cartesian product:

$A\times B=\{(a,b):a\in A,\ b\in B\}$

If $A$ has $p$ elements and $B$ has $q$ elements, $A\times B$ has $pq$ elements. In general $A\times B\mathrel{\htmlClass{course-neq}{\char"2260}} B\times A$.

2.2 Relations

A relation $R$ from $A$ to $B$ is a subset of $A\times B$.

Domain: first components that appear. Range: second components that appear. Codomain: the whole set $B$.

If $(a,b)\in R$ we write $a\,R\,b$.

2.3 Functions

A function $f:A\to B$ is a relation from $A$ to $B$ such that every $a\in A$ is related to exactly one $b\in B$. We write $b=f(a)$.

• Domain: $A$
• Codomain: $B$
• Range: $\{f(a):a\in A\}\subseteq B$

Types (this chapter)

• Identity: $I_A(x)=x$
• Constant: $f(x)=c$
• Polynomial, rational, modulus, signum, greatest integer (as real functions)

Real functions are often given by a formula with an implied domain (all $x$ for which the expression is defined).

Chapter Summary

• $A\times B$ = all ordered pairs $(a,b)$
• Relation = subset of a Cartesian product
• Function: each domain element has exactly one image
• Range $\subseteq$ codomain

Exercises (NCERT)

• Exercise 2.1 — Cartesian products
• Exercise 2.2 — relations
• Exercise 2.3 — functions
• Miscellaneous Exercise