8.1 Equality

One mathematical symbol you are perhaps most familiar with is $=$ . We have been using it fairly freely and, so far, have seen many examples of its use, for example, in Section 2.3.3, we saw $0.1=0.0\overline{9}$ .

If two objects are equal, they are, in some sense, “the same”. This means there is some rule or definition to which we can compare and judge them. We do have to be careful with the objects we choose to make sure they can be compared: we can’t compare the area of a tree with a set.

Often, a new definition will include a rule for when two objects are equal: for example, see the definition of a set or a function in Blocks 2 and 3 respectively. However, again we have to be careful as the context in which we are considering the same objects matters.

Example 8.1.

As algebraic fractions, we can factor and cancel terms in $\frac{x-1}{x^{2}-1}$ to show this is equivalent to $\frac{1}{x+1}$ .

As functions, $\frac{x-1}{x^{2}-1}$ has an implied domain of $\mathbb{R}\setminus\{1,-1\}$ whereas $\frac{1}{x+1}$ has an implied domain of $\mathbb{R}\setminus\{-1\}$ meaning they are different functions.

As you learn more mathematics, you will also come across increasingly general and abstract definitions. These must also be treated with care as they can end up showing two very different looking objects are the “same". For example, famously, to a topologist¹¹ 1 Someone who studies topology: the study of spaces., a mug and a doughnut are equal (you’ll learn more about Homeomorphisms in later years).