Question

Consider ${\sigma }_1=\mathrm{\forall }x(x\sim x),{\sigma }_2=\mathrm{\forall }xy(x\sim y\to y\sim x),{\sigma }_3=\mathrm{\forall }xyz(x\sim$ $y\wedge y\sim z\to x\sim z$ ). Show that if $\mathfrak{Q}\models {\sigma }_1\wedge {\sigma }_2\wedge {\sigma }_3$, where \mathfrak{ } $=⟨A,R)$, then $R$ is an equivalence relation. N.B. $x\sim y$ is a suggestive notation for the atom $\overline{R}(x,y)$.

Short answer

$R$ is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity.

Step-by-step solution

Understanding the Premises

We have three logical statements describing properties of a binary relation $R$ on a set $A$. The expressions use the notation $x\sim y$ to mean $\bar{R}(x,y)$, suggesting that $x$ is related to $y$. We'll check if this makes $R$ an equivalence relation.

Analyzing ${\sigma }_1$

The statement ${\sigma }_1=\mathrm{\forall }x(x\sim x)$ means that for any element $x$ in $A$, $x$ is related to itself. This is the reflexivity condition in equivalence relations.

Analyzing ${\sigma }_2$

${\sigma }_2=\mathrm{\forall }xy(x\sim y\to y\sim x)$ indicates that if $x$ is related to $y$, then $y$ is related to $x$. This describes the symmetry condition of equivalence relations.

Analyzing ${\sigma }_3$

The statement ${\sigma }_3=\mathrm{\forall }xyz(x\sim y\wedge y\sim z\to x\sim z)$ indicates transitivity, which means if $x\sim y$ and $y\sim z$, then $x\sim z$. This is required for $R$ to be an equivalence relation.

Conclusion of Equivalence Relation

If $\mathfrak{Q}\models {\sigma }_1\wedge {\sigma }_2\wedge {\sigma }_3$, then $R$ satisfies the conditions for reflexivity, symmetry, and transitivity. Therefore, $R$ is an equivalence relation.

Key concepts

Logical Statements

Logical statements are expressions that can either be true or false. In mathematics, they form the backbone of reasoning by describing precise conditions. In our case, three logical statements, denoted as ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$, define the properties of a binary relation $R$ on a set $A$. Each statement asserts a condition:

• ${\sigma }_1$ claims that every element is related to itself.
• ${\sigma }_2$ suggests the relationship is mutual between elements.
• ${\sigma }_3$ demands that the relationship can be extended through a third element.

Logical statements not only help in understanding such conditions but also in proving theorems, as they clearly specify what must hold true for a relation to have certain properties. They translate abstract concepts into a form that is easier to reason about and analyze.

Reflexivity

Reflexivity is one of the fundamental properties of equivalence relations. It states that every element in a set is related to itself. In our exercise, this is represented by the logical statement ${\sigma }_1=\mathrm{\forall }x(x\sim x)$. This means "for all $x$, $x$ is related to $x$," ensuring reflexivity:

• Imagine a set $A$ with elements $a,b,c$.
• According to reflexivity, we must have $a\sim a$, $b\sim b$, and $c\sim c$.

In any context where reflexivity holds, you can be sure that each element is self-related. This self-relatedness forms the bedrock upon which symmetry and transitivity operate, ultimately paving the way for a full equivalence relation.

Symmetry

Symmetry in relations means that if one element is related to another, then the second is related back to the first. It ensures mutual relationship between pairs. Our logical statement ${\sigma }_2=\mathrm{\forall }xy(x\sim y\to y\sim x)$ formalizes this concept:

• Suppose you have two elements, say $x$ and $y$.
• Symmetry requires that if $x$ is related to $y$ ($x\sim y$), then $y$ must also be related to $x$ ($y\sim x$).

This property is crucial in many applications, like proving symmetry in geometric figures or ensuring balanced transformations in physics. In an equivalence relation, symmetry complements reflexivity and is necessary for a balanced bidirectional connection.

Transitivity

Transitivity is the property that allows us to "chain" relationships together. If one element is related to a second, and the second is related to a third, transitivity insists that the first is also related to the third. In our example, the logical statement ${\sigma }_3=\mathrm{\forall }xyz(x\sim y\wedge y\sim z\to x\sim z)$ captures this:

• Given three elements $x$, $y$, and $z$, if $x$ is related to $y$ and $y$ to $z$, then $x$ must be related to $z$.
• This ensures a continuous relation chain without breaks.

Transitivity allows more extensive structures within relations, enabling the building of equivalence classes. It's a cornerstone for defining a system's coherence and overall structure, ensuring every connection remains complete and uninterrupted.