MHB Why doesn't it come from a cartesian product of sets?

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The discussion centers on the set defined as $I_A=\{ <a,a>, a \in A \}$, which consists of ordered pairs where both elements are the same, contrasting with the Cartesian product $A \times A$ that includes all possible pairs. It is clarified that $I_A$ does not arise from a Cartesian product when the set $A$ has more than one element, as this would necessitate the inclusion of pairs like $(a,b)$ where $a \neq b$. The reasoning involves showing that if $I_A$ were equal to a Cartesian product $B \times C$, then both $B$ and $C$ would need to contain all elements of $A$, leading to contradictions. The conclusion emphasizes that $I_A$ is not equivalent to any Cartesian product of two sets when $|A|>1$. This distinction is crucial for understanding the nature of relations in set theory.
evinda
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Hello! (Wave)

There is the following sentence in my notes:

Let $A$ be a set. We define the set $I_A=\{ <a,a>, a \in A \}$.

$$A \times A=\{ <a_1,a_2>: a_1 \in A \wedge a_2 \in A \}$$

Then $I_A$ is a relation, but does not come from a cartesian product of sets.

Could you explain me the last sentence? (Thinking)
 
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To illustrate with an example. Let $A = \{a_1,a_2\}$ then $I_A = \{(a_1,a_1),(a_2,a_2)\}$ and $A \times A = \{(a_1,a_1),(a_1,a_2),(a_2,a_1),(a_2,a_2)\}$. Clearly they're not the same and as you mentioned $I_A$ does not come from a cartesian product. Look at how $I_A$ is defined, it only contains the couples where both elements are the same. The cartesian product contains all the possible couples.
 
I still haven't understood why $I_A$ does not come from a cartesian product of sets.. (Sweating)
Could you explain it further to me? (Thinking)
 
evinda said:
I still haven't understood why $I_A$ does not come from a cartesian product of sets..
It apparently means that $I_A$ is not equal to a Cartesian product of two sets if $|A|>1$. Indeed, suppose that $a,b\in A$ and $a\ne b$. Then $\langle a,a\rangle\in I_A$ and $\langle b,b\rangle\in I_A$. Suppose now that $I_A=B\times C$. Then $B$ must contain all first elements of pairs from $I_A$; in particular, $a,b\in B$. Similarly, $C$ contains all second elements of pairs from $I_A$; in particular, $a,b\in B$. But then $\langle a,b\rangle\in B\times C$ even though $\langle a,b\rangle\notin I_A$.
 
Evgeny.Makarov said:
It apparently means that $I_A$ is not equal to a Cartesian product of two sets if $|A|>1$. Indeed, suppose that $a,b\in A$ and $a\ne b$. Then $\langle a,a\rangle\in I_A$ and $\langle b,b\rangle\in I_A$. Suppose now that $I_A=B\times C$. Then $B$ must contain all first elements of pairs from $I_A$; in particular, $a,b\in B$. Similarly, $C$ contains all second elements of pairs from $I_A$; in particular, $a,b\in B$. But then $\langle a,b\rangle\in B\times C$ even though $\langle a,b\rangle\notin I_A$.

I understand... Thank you very much! (Smile)
 
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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