Set

Set is a collection of elements where their order and recurrence does not matter. It could be specified using either 202204281704# or 202204281700#. Due to the properties of set, the set \(\{a, b, c\}\) is same with \(\{b, a, c\}\) and \(\{a, a, b, c, c, c\}\), which could be denoted as \(\{a, b, c\} \equiv \{b, a, c\} \equiv \{a, a, b, c, c, c\}\).

Note: There is a fundamental difference between an element \(e\) and a set that contains only that element \(\{e\}\). (\(e \not = \{e\}\))

If there are too many numbers to express, we can abbreviate them using ellipsis (read “and so forth”) as \(\{1,2,3 \cdots 100\}\). There are some 202204281506# that have their own special symbols.

From the set \(S\), we could see that element \(a\) is an element of \(S\). This could be expressed using \(a \in S\). Let’s say we have an element \(d\), but we know that \(d\) is not an element of \(S\), we can denote it as \(d \not \in S\).

Relations

Sets can have various relations with each other. See the following:

Links to this page
  • Subset

    When all the elements of a #202204281446 \(S\) is also an element in another set \(P\), we could say that \(S\) is a subset of \(P\) which is denoted as \(S \subseteq P\). From this property, we know that \(P\) is also a subset of itself, thus \(P \subseteq P\).

  • Set-Roster Notation

    #202204281446 could be expressed in Set-Roster Notation. Let’s say there is a set that contains element \(a\), \(b\) and \(c\). We could express them as \(S = \{a, b, c\}\).

  • Set-Builder Notation

    #202204281446 could be expressed in Set-Builder Notation like \(\{a \in S | P(x)\}\) which means the set of all elements \(a\) in \(S\) such that \(P(X)\) is true. (\(\{\) and \(\}\) denote the set of all, \(|\) denote such that) This could be written as \(\{a | P(x)\}\).

  • Ring

    Ring is a Set# \(R\), together with two operations \(+\) and \(*\), which has the following properties including those in #Group and #Abelian Group:

  • Relation

    Relation is a #202204281446 of #202204281552 \((a, b)\) where \(a \in A\) and \(b \in B\) assuming that \(A\) and \(B\) are sets, which \(a\) is related to \(b\), denoted \(a R b\). In this sense, \(A\) is the domain of this relation and \(B\) is the co-domain. We could say that it is a relation from A to B.

  • Prime Factorisation
    \(P\) is a Set of prime numbers
  • Ordered Pair

    An ordered pair \((a, b)\) is a pair of objects or a #202204281446 of two sets that is ordered, as defined by Kuratowski \(\{\{a\}, \{a, b\}\}\).

  • Modular Arithmetic

    We could get the next in line congruent residue by Modulo Reduction#. Based on this, we could do addition and multiplication, then modulo reduce the answer when doing Modular Arithmetic with any group of integers (that is Set# \(\mathbb{Z}_n = \{ 0, 1, \ldots, n - 1 \}\)). Additionally, the largest number in Modular Arithmetic (take note on doing addition and multiplication) would be \(n -1\) as it is a clock arithmetic. There are some properties of Modular Arithmetic that need to take note:

  • Group

    Group is a Set# of elements \(S\), together with one operation \(+\), which has the following properties:

  • Galois Field

    Common in Cryptography#, \(GF(p)\) is a Set# where \(\{a \in P | a \in \mathbb{Z} \text{ and } 0 \le a \le p - 1 \}\) with modulo# prime \(p\) as its arithmetic operations. For example, \(GF(7)\) multiplication looks like the table below.

  • Field

    Field is a Set# \(S\), together with two operations \(+\) and \(*\), which has the following properties including those in #Group, #Abelian Group, #Ring, #Communicative Ring, and #Integral Domain:

  • Euler Totient Function

    Euler Totient Function \(\phi(n)\) is defined as number of elements in reduced set of residues. Reduced set of residues is the numbers in a Set of residues \(Z_n = \{0, 1, \ldots, n-1\}\) which are relatively prime# to \(n\). There are several rules on computing Euler Totient Function:

  • Commonly Used Set

    The following are the commonly used #202204281446:

  • Chinese Remainder Theorem (CRT)

    CRT is used to speed up modulo computations by computing smaller moduli \(m_i\) in the Set \(\mod M = m_1 m_2 \ldots m_3\) instead of full modulus \(M\).

  • Cartesian Product

    A Cartesian Product of two #202204281446 \(A\) and \(B\), is the set of all #202204281552 \((a, b)\) where \(a \in A\) and \(b \in B\). It is denoted as \(A \times B\) which read as “A cross B”. This could be expressed in 202204281700# as \(A \times B = \{(a, b) \in S|a \in A \ \text{and}\ b \in B\}\).

  • Average Code-Word Length
    \(k\) is the Set# ${0, 1, \ldots, K - 1}$
#math #logic