1.3 General Terminology: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Equalities and Inequalities
We begin with equality and inequality, because comparing two quantities is one of the most basic operations in mathematics.
We read an equation such as
[latex]a=b,[/latex]
as a equals b, or a is equal to b.
Whereas, if we write
[latex]a \neq b,[/latex]
we say a is not equal to b.
The symbol [latex]\approx[/latex] expresses approximate equality. For example,
[latex]a \approx b,[/latex]
is read as a is approximately equal to b.
If [latex]a \neq b[/latex], and we want to be more specific on which of the two quantities is larger and which is smaller, and hence, we use inequalities.
For example,
[latex]a>b,[/latex]
is read as a is greater/bigger/larger than b.
If we have instead
[latex]a \geq b,[/latex]
this is read as a is greater than or equal to b.
On the other hand,
\[a<b,\]
is read as a is less/smaller than b,
and similarly,
[latex]a \leq b,[/latex]
is read as a is less than or equal to b.
Language Focus: equality and inequality
The word inequality is formed from the prefix in- (meaning not) and the noun equality. It therefore means, literally, the state of not being equal. In everyday language, inequality is often used in the context of social or economic inequality.
In mathematics, however, we use inequality to denote that that one expression is greater than or less than another, using one of the four ordering symbols \( <\), [latex]\leq[/latex], [latex]>[/latex], or [latex]\geq[/latex]. For example, [latex]a > b[/latex] is an inequality.
By contrast, a statement like [latex]a \neq b[/latex] simply tells us that [latex]a[/latex] and [latex]b[/latex] are different; it does not say which one is larger. Hence, [latex]a \neq b[/latex] implies either [latex]a > b[/latex] or \( a< b\); however, conventionally, we do not refer to [latex]a \neq b[/latex] itself as an inequality. The term inequality is reserved for statements involving one of the four ordering symbols above.
Sets
Once we can compare numbers, the next step is to talk about collections of numbers or objects, which we call sets. A set is a collection of objects in which order has no significance. The objects in a set are called its elements. If an element [latex]a[/latex] belongs to a set [latex]A[/latex] we write
\[a \in A, \]
and say a is an element of A, or a belongs to A .
The empty set is the set containing no elements, written
\[ \emptyset, \]
and read the empty set.
If every element of a set is also an element of a set [latex]B[/latex], we say that is a subset of and write
\[ A \subseteq B, \]
which we read as A is a subset of B.
\[ A \subsetneq B, \]
which is read out as A is a proper subset of B.
Conversely, if is a subset of , then is a superset of [latex]B[/latex]. We can write
\[ A \supseteq B, \]
and say A is a superset of B.
The three standard operations on sets are the union [latex]\cup[/latex], intersection [latex]\cap[/latex], and complement [latex]A^c[/latex].
\[ A \cup B, \]
is read out as the A union B. This is the set of elements that are in [latex]A[/latex] or [latex]B[/latex] (or both).
\[ A \cap B, \]
is read out as the A intersection B. This is the set of elements belonging that are both in [latex]A[/latex] and [latex]B[/latex].
Finally,
\[ A^c, \]
is read out as the complement of A or A complement. This is the set of all elements not in [latex]A[/latex].
Mathematics Focus
The complement is always taken with respect to a universal set, which we may write as [latex]\mathcal{E}[/latex], [latex]\mathcal{U}[/latex] or [latex]U[/latex]. For example, if the universal set is the set of all integer numbers and [latex]A[/latex] is the set of even numbers, then [latex]A^c[/latex] is the set of all odd numbers. See the Mathematics Focus in Section 5.3 for more details on the different sets of numbers.
Other notations for the complement include [latex]A^{\prime}[/latex] and [latex]\overline{A}[/latex].
When we work with real numbers, one particularly important kind of set is an interval on the number line. An interval, as a special type of set: it is a connected portion of the real line. If the endpoints [latex]a[/latex] and [latex]b[/latex] are finite and are included, we call
\[ [a, b], \]
a closed interval, which we read out as the closed interval a b or the interval from a to b inclusive.
If the endpoints are not included, we write
\[ (a, b), \]
and call this an open interval which is read out as the open interval a b.
Mathematics Focus
In your studies, you will also meet the same brackets used in a different way when working in two or three dimensions.
The notation [latex](a,b)[/latex] is also used for an ordered pair in two dimensions, and [latex](x,y,z)[/latex] for an ordered triple in three dimensions. The meaning is determined by the context: if we say “the point (x,y) in the plane”, we are not talking about an interval. It is useful to be aware of this double use, so that you do not confuse intervals with coordinates later on. See Section 6.3 for more details of ordered pairs.
Logic
To write proofs and definitions clearly, mathematicians need a precise language for reasoning. A logical proposition is a statement that is either true or false (but not both at the same time).
We combine propositions using the logical operators, AND ([latex]\land[/latex]), OR ([latex]\lor[/latex]) and NOT ([latex]\neg[/latex]):
\[ p \land q, \]
and read p and q.
\[ p \lor q, \]
and this is read out as p or q.
\[ \neg p, \]
and read not p, or the negation of p.
Language Focus
In everyday English, 'or' is sometimes exclusive, as in 'You can have tea or coffee' (but not both). In mathematics, unless the context says otherwise, ' or is inclusive, that is, it allows the possibility that both and are true.
The implication ([latex]\rightarrow[/latex]) expresses that if the first proposition is true, then the second must also be true
\[ p \rightarrow q. \]
This is read out as p implies q, or if p then q.
Whereas if and only if ([latex]\leftrightarrow[/latex]) expresses a two-way connection, which is written as
\[ p \leftrightarrow q, \]
which is read out as p if and only if q.
\[ \forall x \, p(x), \]
which is read out as for all x p of x or for every , p of x holds.
\[ \exists x \, p(x), \]
which is read out as there exists x such that p of x, or there exists x with p of x.
Mathematics Focus: without loss of generality
Some phrases appear so often in proofs that they have become part of the standard 'grammar' of mathematical writing.
For example, suppose we wish to prove a result about any two distinct real numbers [latex]a[/latex] and [latex]b[/latex]. We do not know in advance which is larger, but one of them must since they are distinct. There is therefore no loss of generality in labelling the larger one and the smaller one ; the other case is obtained by swapping the labels.
Rather than writing the same proof out twice, once assuming [latex]a > b[/latex] and once assuming [latex]b > a[/latex], we write it once and begin with: 'without loss of generality, assume [latex]a > b[/latex]'. This signals to the reader that we are making a convenient choice of labels and that the other case works by the same argument.
This is just one example where the phrase without loss of generality can be used in mathematics.
Function Notation
Functions provide a way to describe how one quantity depends on another, so the language of functions appears throughout STEM.
A function [latex]f[/latex] is a rule that assigns to each input [latex]x \in D[/latex] exactly one output value [latex]y=f(x)[/latex]. The variable [latex]x[/latex] is called the independent variable and the dependent variable. The set [latex]D[/latex] is the domain of the set of all output values [latex]f(D)[/latex] is called the range of [latex]f[/latex].
We read [latex]f(x)[/latex] as f of x.
An explicit function is a function where the output is written directly in terms of the input. For example
\[f(x)=\sin (x)\]
is an explicitly defined function and this is read out as f of x is equal to sine of x.
This is read out as x squared y plus, tan of y is equal, to e to the power of, x y.
In such cases we may not be able to solve the equation to write [latex]y[/latex] explicitly as a function of [latex]x[/latex], but we can still treat the relationship as defining an implicit function.
Given a function [latex]f(x)[/latex], its inverse function [latex]f^{-1}(x)[/latex] (if it exists) reverses the effect of [latex]f[/latex]. We read [latex]f^{-1}(x)[/latex] as f inverse of x.
It is important not to confuse the notation the inverse function with the notation of the reciprocal function. Fractions and reciprocals are fundamental in all mathematics, so it is important to be confident with how we say them. The phrase 'over' is commonly used to describe the denominator of a fraction, for example 'a over b'. Finding the reciprocal of a fraction means interchanging its numerator and denominator. Hence, the reciprocal of a non-zero number [latex]a[/latex] is simply 'one over [latex]a[/latex]', a key step in dividing fractions and simplifying algebraic expressions.
Note that the inverse function [latex]f^{-1}(x)[/latex] is not the same as the reciprocal of [latex]f (x)[/latex]. The reciprocal of [latex]f (x)[/latex] is written as
\[\frac{1}{f(x)},\]
and is read out as one over f of x.
Mathematics Focus
Note the different notation for distinguishing
- the inverse [latex]f^{-1}(x)[/latex], and
- the reciprocal [latex](f(x))^{-1}[/latex] which is \(\frac{1}{f(x)}\).
Given two functions [latex]f[/latex] and [latex]g[/latex], the composite function [latex]f \circ g[/latex] is defined by:
\[(f \circ g)(x)=f(g(x)).\]
This is read as f composed with g, of x is equal to, f of g of x.
Arithmetic Operations
Once you are familiar with functions, the next natural question is how to add, subtract, multiply, and divide them.
The four arithmetic operations, addition, subtraction, multiplication, division, can be applied to functions as well as to numbers.
Given two functions [latex]f(x)[/latex] and [latex]g(x)[/latex], we denote the sum of [latex]f(x)[/latex] and [latex]g(x)[/latex] by
\[f(x)+g(x).\]
This is read out as f of x plus g of x, or the sum of f of x and g of x.
On the other hand we denote the difference of [latex]f(x)[/latex] and [latex]g(x)[/latex] by
\[f(x)-g(x),\]
which is read out as f of x minus g of x.
Mathematics Focus
Furthermore, we denote the product of [latex]f(x)[/latex] and [latex]g(x)[/latex] by
\[f(x) \cdot g(x).\]
This is read out as f of x times g of x, or f of x multiplied by g of x, or the product of f of x and g of x.
Finally, we write the quotient of [latex]f(x)[/latex] and [latex]g(x)[/latex] as
\[\frac{f(x)} {g(x)}.\]
We read this as f of x divided by g of x, or f of x over g of x.
Mathematics Focus: Monotonicity
To understand the shape of a graph, we often want to know whether a function is going up, going down, or staying flat.
Let [latex]x_1, x_2 \in I[/latex] with [latex]x_1 \le x_2[/latex].
A function [latex]f(x)[/latex] is:
- increasing on [latex]I[/latex] if [latex]f(x_1) \le f(x_2)[/latex], and
- strictly increasing on [latex]I[/latex] if \( f(x_1) < f(x_2)\).
Similarly, [latex]f(x)[/latex] is:
- decreasing on [latex]I[/latex] if [latex]f(x_1)\geq f(x_2)[/latex], and
- strictly decreasing on [latex]I[/latex] if [latex]f(x_1)>f(x_2)[/latex].
Note: in this book, increasing allows equality, so a function may be flat on parts of the interval and still count as increasing. Some textbooks use increasing to mean strictly increasing. Always check the convention in your module.
Specific functions
Many common functions have special names, and learning this vocabulary will help you recognise them quickly in examples and exam questions.
A polynomial [latex]p(x)[/latex] has the form
\[p(x)=a_n x^n+a_{n-1} x^{n-1}+ \cdots + a_1 x+a_0,\]
which is read out as p of x is equal to, a n x to the n plus, a n minus one x to the n minus one plus, dot dot dot plus, a one x plus a zero.
Here, [latex]a_0, a_1, \cdots, a_{n-1}, a_n[/latex] are the coefficients, [latex]n[/latex] is the degree of the polynomial.
An [latex]n[/latex]-th polynomial has at most [latex]n[/latex] real zeros or roots, i.e., values of [latex]x[/latex] at which [latex]p(x)=0[/latex].
Language Focus
A polynomial of degree two is called a quadratic and a polynomial of degree three is called a cubic. For more details see the Language Focus 3 in Section 1.6.
A rational function, is a quotient of two polynomials [latex]p(x)[/latex] and [latex]q(x)[/latex]:
\[\frac{p(x)}{q(x)}.\]
See Section 3.3 for more details on the integration of rational functions. Furthermore, see the Mathematics Focus in Section 5.3 for more details on rational numbers.
An exponential function has the form
\[f(x)=b^x,\]
which is read out as f of x is equal to, b to the power x or simply, b to the x.
Here [latex]b[/latex] is a constant and is called the base, and [latex]x[/latex] is the variable and is called the exponent. When, [latex]b=e[/latex] (Euler's number), we obtain the exponential function [latex]e^x[/latex] which is central in calculus.
A logarithmic function is the inverse of an exponential function. For base [latex]b[/latex] we write
\[f(x)=\log_b (x),\]
which is read as f of x is equal to log base b of x.
When [latex]b=e[/latex], it is called the natural logarithm, denoted by [latex]\ln(x)[/latex]. This is read out as l n of x
For a non-negative integer [latex]n[/latex], the factorial of [latex]n[/latex] is defined as
\[n!=n (n-1) \cdots 2 \cdot 1, \text{ for } n\geq 1.\]
We read [latex]n![/latex] as n factorial.
See Section 4.3 for more details on how the factorial function is used in Taylor series.
The basic trigonometric functions are [latex]\sin \theta[/latex], [latex]\cos \theta[/latex], and [latex]\tan \theta[/latex]. These are read out as sine theta, cos theta and tan theta.
Their reciprocals are [latex]\csc \theta[/latex], [latex]\sec \theta[/latex], and [latex]\cot \theta[/latex], respectively. These are read out as cosec theta, sec theta and cot theta.
Do not confuse these with the inverse trigonometric functions [latex]\arcsin \theta[/latex], [latex]\arccos \theta[/latex], and [latex]\arctan \theta[/latex]. These are read out as arc sine theta, arc cos theta and arc tan theta.
The trigonometric functions are linked to the hyperbolic functions. See Section 5.3 for more details.
A piecewise function has different definitions on different parts of its domain. For example, the absolute value function is
\[ |x| = \begin{cases} \phantom{-} x & x \geq 0 \\ -x & x < 0 \end{cases}\, \, ,\]
which is read out as the absolute value of x is equal to, x if x is greater than or equal to zero, and it is equal to minus x if x is less than zero.
If the parts of a piecewise function are all constants, then the function is called a step function, because its graph looks like steps.
A function [latex]f(x)[/latex] is said to be an even function if
\[f(-x)=f(x).\]
Whereas a function [latex]f(x)[/latex] is said to be an odd function if
\[f(-x)=-f(x).\]
Mathematics Focus
Do not confuse even and odd functions with even and odd integers: an even number is divisible by two, and an odd number is not. Why are the same terms used?
The terminology comes from the behaviour of power functions [latex]x^n[/latex]:
-
If [latex]n[/latex] is an even number (such as [latex]x^2[/latex] or [latex]x^4[/latex]) the function satisfies [latex]f(-x) = f(x)[/latex], so it is called an even function.
-
If [latex]n[/latex] is an odd number (such as [latex]x^3[/latex] or [latex]x^5[/latex]) the function satisfies [latex]f(-x) = -f(x)[/latex], so it is called an odd function.
The names even function and odd function were extended from these special cases to all functions satisfying the same symmetry properties.
A recursion formula (or recurrence relation) is a mathematical relationship that expresses a function [latex]f_n[/latex] in terms of earlier terms [latex]f_i[/latex] with \( i < n\). For example,
\[ \cos\, (n x) = 2 \cos\, ((n-1)\,x)\cos\, x \, - \,\cos \,((n-2) \, x),\]
gives [latex]\cos\, (n x)[/latex] in terms of [latex]\cos\, ((n-1)\,x)[/latex] and [latex]\cos \,((n-2) \, x)[/latex].
Language Focus
Another term for the same concept as recursion formula is recurrence relation. The prefix re- means 'again' or 'back', and this fits the idea of recursion precisely, where each new term is defined by referring back to earlier ones. You will see re- in many other mathematical terms, such as rearrange, rewrite, and restate, where the idea of doing something again or in reverse is equally clear.
To study change and motion in STEM, we need a way to talk about what happens as a variable gets close to a particular value without necessarily reaching it.
A limit is a mathematical tool that allows us to describe the behaviour of a function [latex]f(x)[/latex] as [latex]x[/latex] gets close to a certain value. We write
\[\lim_{x \to c} f(x), \]
which is read as the limit of f of x, as x tends to/approaches c. Note that one can change the order, and read this out as the limit as x tends to/approaches c, of f of x. The action is called taking the limit.
Note that we can also take one-sided limits, approaching from the right or from the left:
\[\lim_{x \to c^+} \sin(x) \quad \text{ and } \quad \lim_{x \to c^-} \cos(x).\]
We read these as the limit of sine of x as x tends to c from the right/above and the limit of cos of x as x tends to c from the left/below, respectively.
We can also take the limit as [latex]x[/latex] tends to positive/negative infinity. For example
\[\lim_{x \to +\infty} e^{x} \quad \text{ and } \quad \lim_{x \to -\infty} e^{x} ,\]
which are read out as the limit of e to the x, as x tends to positive infinity and the limit of e to the x, as x tends to negative infinity.
Sometimes the function values grow arbitrarily large near a point [latex]a[/latex]. If [latex]f(x)[/latex] becomes arbitrarily large and positive as [latex]x[/latex] approaches [latex]a[/latex], we write
\[\lim_{x \to a} f(x) = \infty, \]
which is read as the limit of f of x as x tends to/approaches a is equal to infinity.
If instead [latex]f(x)[/latex] becomes arbitrarily large and negative, we write
\[\lim_{x \to a} f(x) = - \infty. \]
This is read out as the limit of f of x as x tends to/approaches a is equal to negative infinity.
A function [latex]f[/latex] is continuous at a point [latex]a[/latex] if
\[\lim_{x \to a} f(x) = f(a). \]
If [latex]f[/latex] is not continuous at [latex]a[/latex], then [latex]a[/latex] is called a point of discontinuity.
Finally, we turn to the Greek alphabet that sits quietly behind much of modern mathematics. Greek letters are widely used in mathematics, and as such, it is important to recognise both the symbols and how to say them.
Here is a list of some commonly used lower-case Greek letters.
| Symbol | Name | Pronunciation | Common mathematical use |
|---|---|---|---|
| [latex]\alpha[/latex] | alpha |
|
angles, coefficients, significance level in statistics |
| [latex]\beta[/latex] | beta |
|
angles, coefficients, beta function |
| [latex]\gamma[/latex] | gamma |
|
Euler–Mascheroni constant, angles |
| [latex]\delta[/latex] | delta |
|
small change, epsilon-delta definition of limits |
| [latex]\varepsilon[/latex] | epsilon |
|
small positive quantity, epsilon-delta proofs |
| [latex]\zeta[/latex] | zeta |
|
Riemann zeta function |
| [latex]\eta[/latex] | eta |
|
efficiency, Dirichlet eta function |
| [latex]\theta[/latex] | theta |
|
angles, polar coordinates |
| [latex]\iota[/latex] | iota |
|
rarely used; occasionally an index |
| [latex]\kappa[/latex] | kappa |
|
curvature |
| [latex]\lambda[/latex] | lambda |
|
eigenvalues, wavelength, scaling factor |
| [latex]\mu[/latex] | mu |
|
mean, measure, Möbius function |
| [latex]\nu[/latex] | nu |
|
degrees of freedom, frequency |
| [latex]\xi[/latex] | xi |
|
variables, random variables |
| [latex]\pi[/latex] | pi |
|
the constant [latex]\pi \approx 3.14159[/latex] |
| [latex]\rho[/latex] | rho |
|
correlation coefficient, radius in spherical coordinates |
| [latex]\sigma[/latex] | sigma |
|
standard deviation, summation operator (lower case) |
| [latex]\tau[/latex] | tau |
|
torque, time constant, occasionally [latex]2\pi[/latex] |
| [latex]\upsilon[/latex] | upsilon |
|
rarely used in pure mathematics |
| [latex]\phi[/latex] | phi |
|
golden ratio, angles, potential functions |
| [latex]\chi[/latex] | chi |
|
chi-squared distribution, characteristic function |
| [latex]\psi[/latex] | psi |
|
wave functions, polygamma function |
| [latex]\omega[/latex] | omega |
|
angular frequency, roots of unity |
Here is a list of some commonly used upper-case Greek letters in mathematics.
| Symbol | Name | Pronunciation | Common mathematical use |
|---|---|---|---|
| [latex]\Gamma[/latex] | Gamma |
|
gamma function [latex]\Gamma(n)[/latex] |
| [latex]\Delta[/latex] | Delta |
|
change or difference, e.g. [latex]\Delta x[/latex]; discriminant |
| [latex]\Theta[/latex] | Theta |
|
asymptotic notation in computer science |
| [latex]\Lambda[/latex] | Lambda |
|
diagonal matrix of eigenvalues |
| [latex]\Xi[/latex] | Xi |
|
occasionally used for sets or functions |
| [latex]\Pi[/latex] | Pi |
|
product notation [latex]\prod[/latex] |
| [latex]\Sigma[/latex] | Sigma |
|
summation notation [latex]\sum[/latex] |
| [latex]\Phi[/latex] | Phi |
|
cumulative normal distribution function |
| [latex]\Psi[/latex] | Psi |
|
wave functions, polygamma function |
| [latex]\Omega[/latex] | Omega |
|
sample space in probability; asymptotic notation |
Learning the names and pronunciations of these letters will help you follow lectures and read textbooks more confidently, especially when different languages pronounce them differently.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
