Example of Set for Relation Restriction to A

In summary, the restriction of a relation $R$ to a set $A$ is the subset of $R$ that only includes elements where the first element is in $A$. For a relation $R$ and a set $A$, the domain of the restriction is the intersection of the domain of $R$ and $A$. Examples of such sets include the restriction of a function from $\mathbb{R}$ to $\mathbb{R}$ defined by $xRy$ if $y=x^2$, the restriction of a relation where $mRn$ if $m$ divides $n$, and the restriction of a relation where $mRn$ if $n^2=m$.
  • #1
evinda
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Hello! (Wave)

Let $R$ be a relation and $A$ a set.
The restriction of $R$ to $A$ is the set:

$$R\restriction A=\{ <x,y>: x \in A \wedge <x,y> \in R\}=\{ <x,y>: x \in A \wedge xRy\}$$

For a relation $R$ and a set $A$, it stands that:

$$dom(R \restriction A)=dom(R) \cap A$$

Could you give me an example of such a set, so that I can see that the above relation stands? (Thinking)
 
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  • #2
If $R$ is a function from $\Bbb R$ to $\Bbb R$ defined by $xRy\iff y=x^2$, then $R\restriction\Bbb N$ is the restriction of function $R$ to $\Bbb N$ in this sense. Here $\operatorname{dom} R=\Bbb R$ and $\operatorname{dom}(R\restriction\Bbb N)=\Bbb R\cap\Bbb N=\Bbb N$.

If $mRn\iff m\text{ divides }n$ where $m,n\in\Bbb N$, then $R\restriction\{2\}=\{\langle2,n\rangle\mid n\text{ is even}\}$. Here $\operatorname{dom} R=\Bbb N$ and $\operatorname{dom}(R\restriction\{2\})=\Bbb N\cap\{2\}=\{2\}$.

If $mRn\iff n^2=m$ where $m,n\in\Bbb N$, then $\operatorname{dom} R$ is the set $\Bbb S$ of all square numbers. Let $\Bbb P=\{2^n\mid n\in\Bbb N\}$. Then $R\restriction\Bbb P=\{\langle 2^{2n},2^n\rangle\mid n\in\Bbb N\}$ and $\operatorname{dom}(R\restriction\Bbb P)=\Bbb S\cap\Bbb P=\{2^{2n}\mid n\in\Bbb N\}$.
 

FAQ: Example of Set for Relation Restriction to A

What is a set in relation restriction to A?

A set in relation restriction to A is a subset of a relation R between two sets A and B. It contains all the elements of R that have an input from set A and an output from set B.

What is the purpose of using set for relation restriction to A?

The purpose of using set for relation restriction to A is to limit the elements of a relation R to only those that have an input from set A. This allows for a more specific and focused analysis of the relation.

How is a set for relation restriction to A represented?

A set for relation restriction to A is typically represented using set-builder notation, such as {x ∈ A | xRy}, where x is an element from set A and y is an element from set B. This notation specifies that the set contains all elements from R that have an input from A.

Can a set for relation restriction to A be empty?

Yes, it is possible for a set for relation restriction to A to be empty. This would occur if there are no elements in R that have an input from set A.

How is a set for relation restriction to A different from a regular set?

A regular set contains a collection of unrelated elements, while a set for relation restriction to A contains elements from a specific relation R that have an input from set A. In other words, a set for relation restriction to A has a specific relationship between its elements, while a regular set does not.

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