Relation and Function
Let \[A=\{1,\,2,\,3,4,\}\], \[B=\{2,\,3\}\]
\[A\times B=\{1,\,2,3,\,4,\}\times \{2,3\}=\{(1,2),(2,2),(3,2),(4,2),(1,3),(2,3),(3,3),(4,3)\}\]
Let we choose an arbitrary set:
\[R=[(1,2),(2,2),(1,3),(4,3)]\]
Then R is said to be the relation between a set A to B.
Definition
Relation R is the subset of the Cartesian Product\[A\times B\]. It is represented as \[R=\{(x,y):x\in A\,\] and \[y\in B\}\] {the 2nd element in the ordered pair (x, y) is the image of 1st element}
Sometimes, it is said that a relation on the set A means the all members / elements of the relation
R be the elements / members of \[A\text{ }\times \text{ }A\].
e.g. Let \[A=\{1,\,2,\,3\}\] and a relation R is defined as \[R=\{(x,y):x<y\] where \[x,y\in A\}\]
Sol. \[\because \]\[\mathbf{A=\{1,}\,\mathbf{2,}\,\mathbf{3\}}\]
\[A\times A=\{(1,1),(2,2),(3,3),(2,1),(3,1),(1,2),(3,2),(1,3),(2,3)\}\]
\[\because \,\,\,\,R=\,\,\,\because x<y\]
\[\because \,\,\,\,R=\{(x,y):x<y,and\,x,y\in A\}=\{(1,2),(2,3),(1,3)\]
Note: Let a set A has m elements and set B has n elements. Then \[n(A\times B)\] be \[m\times n\]elements so, total number of relation from A to B or between A and B be\[{{2}^{m\times n}}\].
- A relation can be represented algebraically either by Roster method or set builder method.
Types of Relation
- Void Relation: A relation R from A to B is a null set, then R is said to be void or empty relation.
- Universal Relation: A relation on a set A is said to be universal relation, if each element of A is related to or associated with every element of A.
- Identity Relation: A relation \[{{I}_{x}}\{(x,x):x\in A\}\] on a set A is said to be identity relation on A.
- Reflexive relation: A relation Ron the set A is said to be the reflexive relation. If each and every element of set A is associated to itself. Hence, R is reflexive iff \[(a,\,a)\in R\,\,\forall \,\,a\in A\].
e.g. \[\Rightarrow \,\,A=\{1,\,2,\,3,\,4\}\]
\[R=\{(1,\,3),(1,1),(2,3),(3,2),(2,2),(3,1),(3,3),(4,4)\}\]is a reflexive relation on f.
Sol. Yes, because each and every element of A is related to itself in R.
- Symmetric relation: A relation R on a set A is said to be symmetric relation iff. \[(x,y)\in R\Rightarrow (y,x)\in R\,\,\forall \,\,x,y\in A\]
i.e. \[x\,R\,y\Rightarrow y\,R\,x\,\,\forall \,\,x,y\in A\]
\[\because \] xRy is read as x is R-related to y.
- Anti-symmetric relation: A relation which is not symmetric is said to be anti-symmetric relation.
- Transitive relation: Let A be any non-empty set. A relation R on set A is said to be transitive relation R iff \[(x,y)\in R\] and \[(y,z)\in R\] then \[(x,z)\in R\,\,\forall \,\,x,y,z\in R.\]
i.e. \[xRy\] and \[yRz\Rightarrow xRz\,\,\forall \,\,x,\,y,\,z\in R\].
Let \[\mathbf{A=\{1,}\,\mathbf{2,}\,\mathbf{3,}\,\mathbf{4\}}\]
\[A\times A=\{(1,,1),(2,1),(3,1),(4,1),(1,2),(2,2),(3,2),(4,2),(1,3),(2,3),(3,3),(4,3),(1,4),(2,4),(3,4),(4,4),\]
\[{{R}_{1}}=(1,1),(2,2),(3,2),(2,3),(3,3),(4,4)\]
\[{{R}_{2}}=(2,2),(1,3),(3,3),(3,1),(1,1)\]
\[{{R}_{3}}=(1,1),(2,2),(3,4),(3,3),(4,4)\]
State about\[{{\mathbf{R}}_{\mathbf{1}}}\], \[{{\mathbf{R}}_{\mathbf{2}}}\] and\[{{\mathbf{R}}_{\mathbf{3}}}\]. Are they reflexive, symmetric, anti-symmetric or transitive relations?
Sol. \[{{R}_{1}}\] is symmetric as well as transitive relation for \[{{R}_{2}}\].
\[{{R}_{2}}\]is not reflexive because \[(4,4)\in
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