MHB Example of Set for Relation Restriction to A

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The discussion focuses on the concept of restricting a relation R to a set A, defined mathematically as R|A = {<x,y>: x ∈ A and <x,y> ∈ R}. Examples illustrate this concept, such as the relation defined by xRy if y = x^2, where the restriction to the natural numbers results in a domain of natural numbers. Another example involves the relation mRn if m divides n, showing that the restriction to the set {2} yields pairs where n is even. Additionally, the relation mRn if n^2 = m is examined, with its restriction to the set of powers of 2 resulting in a specific domain of square numbers. These examples clarify the relationship between the domain of the original relation and its restriction.
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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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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 $\operatorname{dom}(R\restriction\Bbb P)=\Bbb S\cap\Bbb P=\{2^{2n}\mid n\in\Bbb N\}$.
 
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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