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Subsection 0.3.1 More on Sets

So we now know how to say that one set is contained within another. We will now define some other operations on sets. Let us also start to be a bit more precise with our definitions and set them out carefully as we get deeper into the text.

Definition 0.3.6

Let \(A\) and \(B\) be sets. We define the union of \(A\) and \(B\text{,}\) denoted \(A \cup B\text{,}\) to be the set of all elements that are in at least one of \(A\) or \(B\text{.}\)

\begin{align*} A \cup B &= \{x | x\in A \mbox{ or } x \in B \} \end{align*}

It is important to realise that we are using the word “or” in a careful mathematical sense. We mean that \(x\) belongs to \(A\) or \(x\) belongs to \(B\) or both. Whereas in normal every-day English “or” is often used to be “exclusive or” — \(A\) or \(B\) but not both  4 When you are asked for your dining preferences on a long flight you are usually asked something like “Chicken or beef?” — you get one or the other, but not both. Unless you are way at the back near the toilets in which case you will be presented with which ever meal was less popular. Probably fish..

We also start the definition by announcing “Definition” so that the reader knows “We are about to define something important”. We should also make sure that everything is (reasonably) self-contained — we are not assuming the reader already knows \(A\) and \(B\) are sets.

It is vital that we make our definitions clear otherwise anything we do with the definitions will be very difficult to follow. As writers we must try to be nice to our readers  5 If you are finding this text difficult to follow then please complain to us authors and we will do our best to improve it..

Definition 0.3.7

Let \(A\) and \(B\) be sets. We define the intersection of \(A\) and \(B\text{,}\) denoted \(A \cap B\text{,}\) to be the set of elements that belong to both \(A\) and \(B\text{.}\)

\begin{align*} A \cap B &= \{ x \;|\; x\in A \mbox{ and } x \in B \} \end{align*}

Again note that we are using the word “and” in a careful mathematical sense (which is pretty close to the usual use in English).

Let \(A = \{1,2,3,4 \}\text{,}\) \(B = \{p : p \mbox{ is prime} \}\text{,}\) \(C = \{5,7,9\}\) and \(D = \{\mbox{even positive integers}\}\text{.}\) Then

\begin{align*} A \cap B &= \{2,3\}\\ B \cap D &= \{2 \}\\ A \cup C &= \{1,2,3,4,5,7,9\}\\ A \cap C &= \es \end{align*}

In this last case we see that the two sets have no elements in common — they are said to be disjoint.

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