2.1 Sets
Before we can start to do maths, we need to know the setting in which we are doing maths. For this reason, sets can be seen as the building block of all maths.
We use curly brackets { and } to denotes sets.
We use the symbol \(\in\) to say an element is in a set. We drawn a line through it to negate it, i.e. we use \(\notin\) to say an element is not in a set.
We use the symbol \(<\) to mean “(strictly) less than” and \(\leq\) to mean “less than or equal to”. Similarly we use \(>\) to mean “greater than” and \(\geq\) to mean “greater than or equal to”The set of trigonometric functions: \(\{\sin(x),\cos(x),\tan(x)\}=\{\cos(x),\tan(x),\cos(x),\sin(x)\}\).
The set of integers between \(2\) and \(6\): \(\{2,3,4,5,6\}=\{6,4,2,2,3,4,6,5\}\).
Since \(2\leq 6\) and \(6\leq 6\), we have \(6 \in \{2,3,4,5,6\}\), however \(15 \notin \{2,3,4,5,6\}\) as \(15>6\).
If a set does not have any elements, we call it the empty set and use \(\emptyset\) (or \(\{\ \}\)).
\(\mathbb{Z}\) is the set of integers, that is \(\mathbb{Z}=\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\).
The symbol \(\mathbb{Z}\) comes from the German “Zahlen” - which means numbers. It was first used by David Hilbert (German mathematician, 1862 - 1943), and popularised in Europe by Nicolas Bourbaki (a collective of mainly French mathematicians, 1935 - ) in their 1947 book “Algèbre”. Integers come from the Latin in, which means “not”, and tag which means “to touch”. An integer is a number that has been “untouched”, i.e. “intact” or “whole”.
The set of integers is equipped with the operations of additions \(+\) and multiplication \(\cdot\) that satisfy the following 10 arithmetic properties (5 relating to addition; a Distributive Law; and 4 relating to multiplication):
(A1) - Closure under addition For all \(x,y \in \mathbb{Z}\) we have \(x+y \in \mathbb{Z}\).
(A2) - Associativity under addition For all \(x,y,z \in \mathbb{Z}\) we have \(x+(y+z)=(x+y)+z\).
(A3) - Commutativity of addition For all \(x,y\in\mathbb{Z}\) we have \(x+y=y+x\).
(A4) - Additive identity For all \(x \in \mathbb{Z}\) we have \(x+0=x\).
(A5) - Additive inverse For all \(x \in \mathbb{Z}\) we have \(-x\in\mathbb{Z}\) and \(x+(-x)=0\).
(A6) - Distributive Law For all \(x,y,z\in\mathbb{Z}\) we have \(x(y+z)=xy+xz\).
(A7) - Closure under multiplication For all \(x,y \in \mathbb{Z}\) we have \(xy \in \mathbb{Z}\).
(A8) - Associativity under multiplication For all \(x,y,z \in \mathbb{Z}\) we have \(x(yz)=(xy)z\).
(A9) - Commutativity of multiplication For all \(x,y\in\mathbb{Z}\) we have \(xy=yx\).
(A10) - Multiplicative identity For all \(x \in \mathbb{Z}\) we have \(1\cdot x=x\).
Formally speaking, this is saying that \(\mathbb{Z}\) with \(+\) and \(\cdot\) is a ring. The notion of a ring is explored in more details in later units such as the second year unit Algebra 2. Properties (A1) to (A5) tells us that \(\mathbb{Z}\) with \(+\) is an abelian group. We will explore the notion of groups and abelian groups later in this unit.
On top of these 10 arithmetic properties, \(\mathbb{Z}\) is well ordered, i.e., it comes with 4 order properties:
(O1) - trichotomy For all \(x,y\in\mathbb{Z}\) either \(x<y\), \(x=y\) or \(x>y\).
(O2) - transitivity For all \(x,y,z\in\mathbb{Z}\), if \(x<y\) and \(y<z\) then \(x<z\).
(O3) - compatibility with addition For all \(x,y,z in \mathbb{Z}\), if \(x<y\) then \(x+z<y+z\).
(O4) - compatibility with multiplication For all \(x,y,z \in \mathbb{Z}\), if \(x<y\) and \(z>0\) then \(zx<zy\).
There are many words above that seems to have come from nowhere, but can be related back to words used in everyday English.
Associative comes from the Latin ad meaning “to” and socius meaning “partner, companion”. An associate is someone who is a companion to you. The property \(x+(y+z)=(x+y)+z\) shows that it doesn’t matter who \(y\) “keeps company with”, the result is still the same.
Commutative comes from the Latin co meaning “with” and mutare meaning “to move”. To commute is “to change, to exchange, to move”. In everyday situation, a commute is the journey (i.e., moving) from home to work. The property \(x+y=y+x\) shows that we can move/exchange \(x\) and \(y\) and the result is the same.
Identity comes from the Latin idem meaning “same”. The additive identity is the element which keeps other elements the same when added to it. The multiplicative identity is the element which keeps other elements the same when multiplied by it.
Inverse comes from in and vertere which is the verb “to turn”. The additive inverse of \(x\) is the quantity that turns back “adding \(x\)”.
Trichotomy comes from the Greek trikha meaning “in three parts” and temnein meaning “to cut”. If you pick \(x\in \mathbb{Z}\), then you can cut \(\mathbb{Z}\) into three parts, the integers less than \(x\), the integers equal to \(x\) and the integers greater than \(x\).
Transitive comes from the Latin trans meaning “across, beyond” and the verb itus/ire meaning “to go”. Knowing \(x<y\) and \(y<z\) allows us to go across/beyond \(y\) to conclude \(x<z\).
As seen from the properties above, we often need to quantify objects in mathematics, that is we need to distinguish between a criteria always being met (“for all”), or the existence of a case where a criteria is met (“there exists”). Sometimes we also need to distinguish whether there is a unique case where a criteria is met.
We have the following symbolic notation:
The symbol \(\forall\) denotes for all, or equivalently, for every.
The symbol \(\exists\) denotes there exists.
We use \(\exists !\) to denote there exists a unique, or equivalently there exists one and only one.
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We can use \(\mathbb{Z}\) as a starting point to construct different sets.
Within the curly brackets of a set, we use colon, \(\colon\), to mean “such that”.
Returning to the previous example, the set of integers between \(2\) and \(6\) can be written as \(\{x \in \mathbb{Z}\colon 2\leq x \leq 6\}\);
We use \(+\) to denote the positive numbers in a set. I.e., \(\mathbb{Z}_+= \{x\in \mathbb{Z}: x> 0\} = \{1,2,\dots,\}\) denotes the set of positive integers.
Similarly, \(\mathbb{Z}_- = \{x\in \mathbb{Z}: x<0 \} = \{-1,-2,-3,\dots\}\) denotes the set of negative integers.
We denote the set of non-negative integers by \(\mathbb{Z}_{\geq 0}=\{x\in \mathbb{Z}: x\geq 0\} = \{0,1,2,\dots,\}\).
If \(n \in \mathbb{Z}\), we write \(n\mathbb{Z} = \{nx:x\in \mathbb{Z}\} = \{y \in \mathbb{Z}: \exists x \in \mathbb{Z} \text{ with } y=xn\} = \{\dots,-3n,-2n,-n,0,n,2n,3n,\dots\}\).
You may have also heard of the natural numbers denoted \(\mathbb{N}\). However, some sources consider \(0\) as a natural number (so \(\mathbb{N} = \mathbb{Z}_{\geq 0}\)) and other consider \(0\) not to be a natural number (so \(\mathbb{N} = \mathbb{Z}_+\)). To avoid any confusion (and because we will need \(\mathbb{Z}_{\geq 0}\) sometimes and \(\mathbb{Z}_+\) at other times), we will not be using \(\mathbb{N}\) in this course.
\(\mathbb{Q}\) is the set of rational numbers, that is \(\mathbb{Q}=\left\{\frac{a}{b}:\ a\in\mathbb{Z},\ b\in\mathbb{Z}_+ \right\}\).
We will later see how \(\mathbb{Q}\) can be constructed from \(\mathbb{Z}\) and how this lead to a “natural” way to write each rational numbers
The symbol \(\mathbb{Q}\) stands for the word “quotient”, which is Latin for “how often/how many”, i.e., the quotient \(\frac{a}{b}\) is “how many times does \(b\) fit in \(a\)”. Surprisingly, the word “rational” to describe some numbers came after the use of “irrational” number. Ratio is Latin for “thinking/reasoning”. When the Phythagorean school in Ancient Greece realised some numbers could not be expressed as the quotient of two whole numbers (such as \(\sqrt{2}\), which we will prove later), they called those “irrational”, i.e. numbers that should not be thought about. “Rational” numbers were numbers that were not “irrational”, i.e., one could think about them.
We extend the operations of addition and multiplication as well as the order relation for the integers to the rational numbers. Let \(\frac{a}{b}, \frac{c}{d} \in \mathbb{Q}\), then:
\[\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd} \text{ and } \frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}.\] Similarly to \(\mathbb{Z}\), we have that \(\mathbb{Q}\) with \(+\) and \(\cdot\) satisfies the properties (A1) to (A10) as well as (O1) to (O4). It also satisfy the extra arithmetic property:
(A11) - multiplicative inverse For all \(x\in\mathbb{Q}\) with \(x\neq 0\), we have \(x^{-1} = \frac{1}{x} \in \mathbb{Q}\) and \(x^{-1}x = 1\).
Notice that (A11) is similar to (A5) but for multiplication. As we will see later in the course, another way of saying (A7) to (A11) is that \(\mathbb{Q}\) without \(0\) under \(\cdot\) is an abelian group. As you will see in Linear Algebra, the arithmetic properties of \(\mathbb{Q}\) ((A1) to (A11)) comes from the fact that \(\mathbb{Q}\) is a field.
Similarly with \(\mathbb{Z}\), using \(\mathbb{Q}\) we can construct the sets \(\mathbb{Q}_+\), \(\mathbb{Q}_-\), \(\mathbb{Q}_{\geq 0}\) etc.