Theory of Positive Integers - Function & Equivalence Relations

Which of the following is a property of a relation?

Transitive Property

Non-symmetric Property

Equivalency Property

Partition Property

All are properties of relations.

Explanation

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

Which of the following is a property of a relation?

Reflexive Property

Non-symmetric Property

Equivalency Property

Associative Property

All are relation properties

Explanation

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

Which of the following is a property of a relation?

Symmetric Property

Non-symmetric Property

Equivalency Property

Partition Property

All are properties of a relation

Explanation

For a relation to exist there must be a non empty set present. If a non empty set is present then there are three relation properties.

These properties are:

I. Reflexive Property

$x\approx x$

II. Symmetric Property

$x\approx y\rightarrow y\approx x$

III. Transitive Property

$x\approx y, y\approx z \rightarrow x\approx z$

When all three properties represent a specific set, then that set is known to have an equivalence relation.

What is an equivalency class?

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A\end{Bmatrix}$

$[x]=\begin{Bmatrix} x\approx y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A, -x\approx y \end{Bmatrix}$

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x>y \end{Bmatrix}$

Explanation

An equivalency class is a definitional term.

Suppose is a non empty set and $\approx$ is an equivalency relation on . Then belonging to is a set that holds all the elements that live in that are equivalent to .

In mathematical terms this looks as follows,

$[x]=\begin{Bmatrix} y\ \epsilon\ A, x\approx y \end{Bmatrix}$

Function & Equivalence Relations

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