One of the most important relations in modern mathematics are equivalence relations. For this important properties must be shown.

Relationship Properties

We start the section with the properties of a relation $\mathcal{R}$ on a set $A$. As a starting point, here is the definition:

Definition Let $\mathcal{R}$ be a relation on a set $A.$ Then we say that:

  • $\mathcal{R}$ is reflexive if the condition:
    • $(\forall x \in A)(x\mathcal{R} x)$ is true
    • if for all $x \in A, (x, x) \in R.$ collection
  • $\mathcal{R}$ is symmetric if the condition:
    • $(\forall x,y \in A)(x\mathcal{R} y \longrightarrow y\mathcal{R} x)$ is true.
    • if for all $x, y \in A,$ if $(x, y) \in \mathcal{R} ,$ then $(y, x) \in \mathcal{R} .$ collection
  • $\mathcal{R}$ is transitive if the condition:
    • $(\forall x, y, z \in A)(x\mathcal{R} y \wedge y\mathcal{R} z \longrightarrow x\mathcal{R} z)$ is true
    • if for all $x, y, z \in A,$ if $(x, y) \in \mathcal{R} $ and $(y, z) \in \mathcal{R},$ then $(x, z) \in \mathcal{R}.$ collection
  • $\mathcal{R}$ is anti-symmetric if the condition:
    • $(\forall x, y \in A)(x\mathcal{R} y \wedge y\mathcal{R} x \longrightarrow x = y)$ is true
    • if for all $x, y \in A,$ if $(x, y) \in \mathcal{R}$ and $(y, x) ∈ \mathcal{R},$ then $x = y.$ collection

Examples

Below we present some examples of relationship properties:

Example 1. Let $A$ be any set. The relationship about $A$:

$$\Delta = \\{(x,x); x\in A\\}$$

Suppose this relation is reflexive, symmetric and transitive. In this case we call it an identity or diagonal relationship.

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Example 2. Let $A$ be any set. The relationship $A\times A$ over $A$.

Suppose this relation is reflexive, symmetric and transitive. In this case it is not anti-symmetric.

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Example 3. Let $A=\\{a,b,c\\}$. The relationship:

  • $\mathcal{R}_1 = \\{(a,a), (b,b), (c,c), (a,b)\\}$ Suppose this relation is reflexive, anti-symmetric and transitive. In this case it is not symmetrical.
  • $\mathcal{R}_2 = \\{(a,a), (b,b), (a,b), (b,a)\\}$ Suppose this relationship is symmetric and transitive. In this case it is neither reflexive nor anti-symmetric.
  • $\mathcal{R}_3 = \\{(a,a), (b,b), (a,b), (b,a), (b,c)\\}$ In this case it is neither symmetric, nor transitive, nor reflexive nor anti-symmetric.
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Example 4. Let $A=\mathbb{N}$. The relationship:

$$\mathcal{R} = \\{(x,y)\in \mathbb{N}\times \mathbb{N}; ~x \text{ é um divisor de } y\\}$$

In this case it is a reflexive, anti-symmetric and transitive relationship. However, it is not symmetrical.

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Example 5. Let $A=\mathbb{Z}$. The relationship:

$$\mathcal{R} = \\{(x,y)\in \mathbb{N}\times \mathbb{N}; ~x-y \text{ é múltiplo de } 3\\}$$

In this case it is a reflexive, symmetric and transitive relationship. However, it is not anti-symmetric.

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Example 6. Let $A$ be a family of sets. For $X, Y \in A,$ the relation “$X$ is contained in $Y$” is a reflexive, anti-symmetric and transitive relation. However, it is not symmetrical.

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Example 7. Let $A$ be the set of propositions. For $p, q \in A,$ the relation “If $p$ then $q$” is a reflexive and transitive relation. It is neither symmetric nor anti-symmetric.

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This is the summary of the properties of relations. Wait for the next posts.