2.2 Definitions

Because mathematical language is precise, it is crucial that you understand exactly what is meant whenever a term or phrase is used.

In mathematical writing, you will often see definitions presented as above, with the term being defined indicated in bold or italics, but definitions can also appear within the body of the text depending on the author’s writing style.

When reading a definition, every word and symbol is important. Consider the following definition.

Note that the symbol ‘…’ is called an ellipsis. In mathematics, it means that the obvious pattern is continued. In the definition above, it should be clear with a moment’s thought what the pattern is. You should be very careful only to use ellipses when the pattern really is obvious and you should be able to give a clear explanation if asked.

Definition 2.3 tells us that, for example:

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  $1$ is a natural number.

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  $1001$ is a natural number.

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  $-1$ is not a natural number; if it were then it would be included as such in the definition.

These may seem like obvious points, but it is very important that you take care to understand definitions precisely. You must not assume that a definition applies to an object just because it is the ‘same sort of thing’ as appears in a definition.

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  Solution.For example: $1\>000\>000$ and $-101$ are integers; $\frac{1}{2}$ and $\pi$ are not.

Definition 2.6 tells us that, for example:

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  $6$ is even because $6=2\times 3$; so $k=3$ in this case.

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  $-10$ is even because $10=2\times(-5)$; $k=-5$ is allowed in the definition since $k$ does not need to be positive.

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  $0$ is even because $0=2\times 0$; $k=0$ is allowed too, $0$ is a perfectly good integer.

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  $1$ is not even because the only $k$ for which $1=2k$ is $k=0.5$, and $0.5$ is not an integer.

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  $-4.2$ is not even because the definition only applies to integers.

Mathematicians introduce definitions so that they can make precise and concise statements. There is no use in proving a statement that uses terms wrongly or that the reader might interpret differently.

Some concepts are so common in mathematics that the mathematical community has standard terms and notation that (usually!) always mean the same thing. For example,

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  the term ‘integer’ always has the meaning from Definition 2.4;

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  the symbol $\pi$ (almost) always means the constant that is the ratio of a circle’s circumference to its diameter.

However, there is no single ‘correct’ way to phrase the definition of a mathematical term. You will likely see the same term defined in different ways throughout your degree programme.

Note that definitions are not mathematical statements: they are not true or false. Rather, definitions provide common ground for further discussion. There can be equivalent definitions and alternative definitions of a mathematical concept. For example, sometimes $0$ is defined to be a natural number and sometimes it is not, as in Definition 2.3.

Beware, too, that definitions sometimes reuse words or phrases in different contexts. For example,

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  the term ‘even’ can also be applied to functions, where it means something different to Definition 2.6;

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  the term ‘graph’ could mean the familiar geometrical object associated with a function, or it could mean a mathematical object consisting of vertices and edges.

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  Solution.Notice that odd integers are of the form $(\text{even number}+1)$, e.g. $1=0+1$, $3=2+1$, and so on.

  Therefore, we could make the definition:

  ‘An integer $n$ is said to be odd if $n=2k+1$ for some integer $k$.’

  There are other ways this definition could be phrased. We could also say:

  ‘An integer $n$ is said to be odd if $n=2k-1$ for some integer $k$.’

  Or we could say:

  ‘An integer $n$ is said to be odd if it is not even.

  These definitions are all equivalent (this is essentially a consequence of Example 4.23). Which is best may depend on the context.

The key things to take away from this section are that:

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  you should always make sure you understand the precise meaning of a term or phrase whenever it is used, as this may be different in different contexts; and

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  you should use examples to help understand what is and is not covered by a definition.