Chapter 1

Relations and Functions

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (18)

Examples

Example 1Empty and universal relations in a boys' schoolworked
Example 2Congruence of triangles is an equivalence relationworked
Example 3Perpendicular lines: symmetric onlyworked
Example 4A relation on {1, 2, 3} that is reflexive onlyworked
Example 5 $2 \mid (a-b)$ is an equivalence relation on $\Z$ worked
Example 6Same parity on {1, …, 7}worked
Example 7Roll numbers: one-one but not ontoworked
Example 8 $f(x)=2x$ on $\N$ ⁠: one-one but not ontoworked
Example 9 $f(x)=2x$ on $\R$ ⁠: bijectiveworked
Example 10Onto but not one-one on $\N$ worked
Example 11 $f(x)=x^2$ on $\R$ ⁠: neitherworked
Example 12Swapping neighbours on $\N$ is bijectiveworked
Example 13Onto on {1, 2, 3} forces one-oneworked
Example 14One-one on {1, 2, 3} forces ontoworked
Example 15Computing $g\circ f$ on finite setsworked
Example 16 $g\circ f \ne f\circ g$ for $\cos x$ and $3x^2$ worked
Example 17Inverse of $f(x)=4x+3$ worked
Example 18The intersection of two equivalence relationsworked
Example 19 $(x,y)\,R\,(u,v) \iff xv=yu$ worked
Example 20Two descriptions of one relation on {1, …, 9}worked
Example 21 $f(a)=f(b)$ defines an equivalence relationworked
Example 22Number of one-one functions on {1, 2, 3}worked
Example 23Reflexive, transitive, not symmetric: countingworked
Example 24Equivalence relations containing (1, 2): countingworked
Example 25 $I_{\N}+I_{\N}$ is not ontoworked
Example 26 $\sin x+\cos x$ is not one-one on $[0,\pi/2]$ worked

Exercise 1.1

Exercise 1.1 · Q1Reflexive, symmetric, transitive: nine relationsworked
Exercise 1.1 · Q2 $a \le b^2$ on $\R$ worked
Exercise 1.1 · Q3 $b = a+1$ on {1, …, 6}worked
Exercise 1.1 · Q4 $a \le b$ on $\R$ worked
Exercise 1.1 · Q5 $a \le b^3$ on $\R$ worked
Exercise 1.1 · Q6{(1, 2), (2, 1)} on {1, 2, 3}worked
Exercise 1.1 · Q7Books with the same number of pagesworked
Exercise 1.1 · Q8 $|a-b|$ even on {1, …, 5}worked
Exercise 1.1 · Q9Multiples of 4, and equality, on {0, …, 12}worked
Exercise 1.1 · Q10Relations with exactly the properties asked forworked
Exercise 1.1 · Q11Points at the same distance from the originworked
Exercise 1.1 · Q12Similar trianglesworked
Exercise 1.1 · Q13Polygons with the same number of sidesworked
Exercise 1.1 · Q14Parallel lines and the class of $y=2x+4$ worked
Exercise 1.1 · Q15 (MCQ)Which properties does this relation on {1, 2, 3, 4} have?worked
Exercise 1.1 · Q16 (MCQ) $a=b-2,\ b>6$ ⁠: which pair belongs?worked

Exercise 1.2

Exercise 1.2 · Q1 $f(x)=1/x$ on the non-zero realsworked
Exercise 1.2 · Q2Injectivity and surjectivity of $x^2$ and $x^3$ worked
Exercise 1.2 · Q3The greatest integer functionworked
Exercise 1.2 · Q4The modulus functionworked
Exercise 1.2 · Q5The signum functionworked
Exercise 1.2 · Q6A function from {1, 2, 3} to {4, 5, 6, 7}worked
Exercise 1.2 · Q7 $3-4x$ and $1+x^2$ on $\R$ worked
Exercise 1.2 · Q8 $(a,b)\mapsto(b,a)$ is bijectiveworked
Exercise 1.2 · Q9Is the halving-and-rounding-up function bijective?worked
Exercise 1.2 · Q10 $f(x)=\frac{x-2}{x-3}$ from $\R-\{3\}$ to $\R-\{1\}$ worked
Exercise 1.2 · Q11 (MCQ) $f(x)=x^4$ on $\R$ worked
Exercise 1.2 · Q12 (MCQ) $f(x)=3x$ on $\R$ worked

Miscellaneous Exercise

Miscellaneous · Q1 $f(x)=\frac{x}{1+|x|}$ is a bijection onto $(-1,1)$ worked
Miscellaneous · Q2 $f(x)=x^3$ is injectiveworked
Miscellaneous · Q3Is $A\subset B$ an equivalence relation on $P(X)$ ⁠?worked
Miscellaneous · Q4Number of onto functions from {1, …, n} to itselfworked
Miscellaneous · Q5Are $x^2-x$ and $2\left|x-\tfrac12\right|-1$ equal?worked
Miscellaneous · Q6 (MCQ)Reflexive and symmetric but not transitive: how many?worked
Miscellaneous · Q7 (MCQ)Equivalence relations containing (1, 2): how many?worked

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