Relation

Abstract

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• Basically a Subset of Cartesian Product or Order n-tuples filtered by some conditions which define the relation among the elements from the given Set

Real-world Implication

Commonly used in Database, the columns are the different sets, the Cartesian Product of the columns are all the potential relation aka all the rows that can be stored inside the database. Each row is a actual Order n-tuples inside the relation

Domain

• Basically the Set of elements of the left hand Set that is involved in the Relation

$$\{a\in A:aRbforsomeb\in B\}$$

Co-domain

• The whole Set at the right hand side

Range

• Basically the Set of elements of the right hand set that is involved in the Relation

$$\{b\in B:aRbforsomea\in A\}$$

Binary Relation

• Let $x\in A,y\in B$, we say $xRy\leftrightarrow (x,y)\in R$

N-ary Relation

• A Relation involving 2 Set is called binary relation, also known as 2-ary
• Ternary is 3-ary
• Quaternary is 4-ary

Inversion of Relation

• Basically reverse the order of elements of the Order n-tuples

$$R^{-1}=\{(y,x)\in B\times A:(x,y)\in R\}$$

Arrow Diagram

• Visualize Relation
• Usually used when there is more than one Set involve in the relation

Theorem

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Theorem 8.3.4

• The Set Partition induced by Equivalence Relation
• If $R$ is equivalence relation on Set $A$, then the distinct Equivalence Class form a partition of $A$

Terminologies

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Transitive Closure of Relation

• The Relation obtained by adding the least number of Ordered Pair to ensure Transitive
• Represented with $R^t$
• Following 3 properties:
  1. $R^t$ is transitive
  2. $R\subseteq R^t$
  3. $R^t\subseteq S$, where $S$ is any other transitive relation that contains $R$