The Real and Complex Number Systems¶
Groups¶
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Symmetry: A symmetry is a bijection from a set to itself, \(s:X\to X\).
Tip
Symmetry is a way of relabelling the items.
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Examples (intuition for symmetries):
The trivial symmetry is the identity map that maps every element of a set onto itself.
A set of 3 points arranged in a circle has a cyclic symmetry (rotate the circle \(1/3\) of a revolution, mapping the points accordingly).
The set of all permutations of a finite set define symmetries, where any element can map to any other element.
\(\forall n \in \mathbb{Z}\) define a symmetry \(x \in \mathbb{Z} \mapsto n + x \in \mathbb{Z}\) (sliding the number line).
\(\forall n \in \mathbb{Q}\) define a symmetry \(x \in \mathbb{Q} \mapsto n \cdot x \in \mathbb{Q}\) (stretching the number line).
For a finite-dimensional vector space over \(\mathbb{R}\), \(\mathbb{R}^n\), the set of non-singular \(n \times n\) matrices define symmetries since it maps any vector \(u \in \mathbb{R}^n \mapsto Au \in \mathbb{R}^n\) (Generalised Linear Group of order \(n\) or \(\mathrm{GL}_n(\mathbb{R})\)).
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Group: A group is a set \(G\) with a binary operation \((+)\) that satisfies:
\(\forall x, y \in G, (x + y) \in G\) [Closure]
\(\forall x, y, z \in G, ((x + y) + z) = (x + (y + z))\) [Associativity]
\(\exists 0 \in G\) such that \(\forall x \in G, (x + 0) = (0 + x) = x\) [Identity Element]
\(\forall x \in G, \exists (-x) \in G\) such that \((x + (-x)) = ((-x) + x) = 0\) [Inverse Element]
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Proposition: For a group \((G, +, 0)\) and \(\forall x, y, z \in G\):
\((x + y = x + z) \Longrightarrow y = z\) [Cancellation Law]
\((x + y) = x \Longrightarrow y = 0\) [Uniqueness of Identity]
\((x + y) = 0 \Longrightarrow y = (-x)\) [Uniqueness of Inverse]
\(-(-x) = x\) [Repeated Inverse]
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Commutative Group (Abelian): A group that follows the additional axiom
\[\forall x, y \in G, (x + y) = (y + x).\]
Fields¶
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Field: A field is a set \(F\) with two binary operations, \((+)\) and \((\cdot)\), which satisfy:
\((F, +, 0)\) is a commutative group [Additive Group]
\((F \setminus \{0\}, \cdot, 1)\) is a commutative group [Multiplicative Group]
\(\forall x, y, z \in F, (x \cdot (y + z)) = ((x \cdot y) + (x \cdot z))\) [Distributive Property]
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Proposition: For a field \((F, +, 0, \cdot, 1)\) and \(\forall x, y \in F\):
\((x \cdot 0) = (0 \cdot x) = 0\)
\((x \ne 0) \wedge (y \ne 0) \Longrightarrow (x \cdot y) \ne 0\)
\((-x) \cdot y = x \cdot (-y) = -(x \cdot y)\)
\((-x) \cdot (-y) = x \cdot y\)
Ordered Sets¶
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Order: For a set \(S\), an order \((<)\) is a relation such that:
For all \(x, y \in S\), exactly one of \(x < y\), \(x = y\), \(y < x\) is true.
For all \(x, y, z \in S\), \((x < y) \wedge (y < z) \Longrightarrow (x < z)\).
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Partial Order: A partial order \((\le)\) has:
\(x \le x\) [Reflexive]
\((x \le y) \wedge (y \le x) \Longrightarrow (x = y)\) [Anti-symmetric]
\((x \le y) \wedge (y \le z) \Longrightarrow (x \le z)\) [Transitive]
Partial order alone is not a total order. To obtain a total order, also require:
\(\forall x, y \in S, (x \le y) \vee (y \le x)\) [Connex]
\(\forall x, y \in S, (x \le y) \Longleftrightarrow \neg (y < x)\).
Note
Ordered Set: A set \(S\) for which an order is defined.
Poset: A set \(S\) for which a partial-order is defined.
Upper Bound: For \(E \subseteq S\), \(\beta \in S\) is an upper bound if \(\forall x \in E, x \le \beta\).
Bounded Above: \(E\) is bounded above if it has an upper bound.
Least Upper Bound: If \(\alpha\) is an upper bound of \(E\) and for every \(\gamma < \alpha\), \(\gamma\) is not an upper bound of \(E\), then \(\alpha = \sup E\).
Greatest Lower Bound / Infimum: Defined similarly for sets bounded below, \(\alpha = \inf E\).
Least Upper Bound Property (Completeness): An ordered set has this property if every non-empty bounded-above set \(E \subseteq S\) has \(\sup E \in S\).
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Theorem: Let \(S\) be an ordered set with the least upper bound property. Let \(B \subseteq S\) be non-empty and bounded below. Let \(L\) be the set of all lower bounds of \(B\). Then there exists \(\alpha \in S\) such that \(\alpha = \sup L = \inf B\). Hence \(\inf B \in S\).
Ordered Field¶
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Ordered Field: A field \(F\) that is also an ordered set, such that:
\(x, y, z \in F,\ y < z \Longrightarrow x + y < x + z\)
\(x, y \in F,\ (x > 0) \wedge (y > 0) \Longrightarrow x \cdot y > 0\)
The Real Field¶
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Real Field: There exists an ordered field \(\mathbb{R}\) with the least upper bound property. Moreover, \(\mathbb{Q} \subset \mathbb{R}\) is a subfield. The elements are called real numbers.
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Theorem (properties of \(\mathbb{R}\) and \(\mathbb{Q}\)):
If \(x, y \in \mathbb{R},\ x > 0\), then \(\exists n \in \mathbb{Z}^{+}\) such that \(n \cdot x > y\) [Archimedean property]
If \(x, y \in \mathbb{R},\ x < y\), then \(\exists p \in \mathbb{Q}\) such that \(x < p < y\) [\(\mathbb{Q}\) dense in \(\mathbb{R}\)]
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Theorem: For \(x \in \mathbb{R}^{+}\) and \(n \in \mathbb{Z}^{+}\), there exists a unique \(y \in \mathbb{R}^{+}\) such that \(y^{n} = x\).
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Corollary: For \(a, b \in \mathbb{R}^{+}\) and \(n \in \mathbb{Z}^{+}\), \((a \cdot b)^{1/n} = a^{1/n} \cdot b^{1/n}\).
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Decimal expansion of reals: For \(x \in \mathbb{R}^{+}\), let \(E\) be the set
\[n_0 + \frac{n_1}{10} + \frac{n_2}{10^{2}} + \cdots + \frac{n_k}{10^{k}}.\]Then \(x = \sup E\).
Extended Real Number System¶
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Extended reals: \(\mathbb{R}\) together with two symbols \(-\infty\) and \(+\infty\), with \(\forall x \in \mathbb{R},\, -\infty < x < +\infty\).
The Complex Field¶
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Complex Numbers: Ordered pairs \((a, b)\) of real numbers, where for \(x = (a, b)\) and \(y = (c, d)\):
\((x = y) \Longleftrightarrow (a = c) \wedge (b = d)\) [Equality]
\(x + y = (a + c, b + d)\) [Addition]
\(x \cdot y = (ac - bd, ad + bc)\) [Multiplication]
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Theorem: \(\mathbb{C}\) is a field with additive identity \((0, 0)\) and multiplicative identity \((1, 0)\).
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Proposition: For \(a, b \in \mathbb{R}\):
\((a, 0) + (b, 0) = (a + b, 0)\)
\((a, 0) \cdot (b, 0) = (ab, 0)\)
Note
Define \(i = (0, 1)\).
Proposition: \(i^{2} = -1\).