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Theory of Positive Integers : Function & Equivalence Relations

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Example Questions

Example Question #1 : Function & Equivalence Relations

Which of the following is a property of a relation?

Possible Answers:

Partition Property

Equivalency Property

All are properties of a relation

Non-symmetric Property

Symmetric Property

Correct answer:

Symmetric Property

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.

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Example Question #1 : Function & Equivalence Relations

What is an equivalency class?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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,

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

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Example Question #2 : Function & Equivalence Relations

Which of the following is a property of a relation?

Possible Answers:

All are relation properties

Associative Property

Non-symmetric Property

Reflexive Property

Equivalency Property

Correct answer:

Reflexive Property

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.

Report an Error

Example Question #3 : Function & Equivalence Relations

Which of the following is a property of a relation?

Possible Answers:

Non-symmetric Property

Equivalency Property

Transitive Property

All are properties of relations.

Partition Property

Correct answer:

Transitive Property

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.

Report an Error

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Theory of Positive Integers : Function & Equivalence Relations

Study concepts, example questions & explanations for Theory of Positive Integers

All Theory of Positive Integers Resources

Example Questions

Example Question #1 : Function & Equivalence Relations

Example Question #1 : Function & Equivalence Relations

Example Question #2 : Function & Equivalence Relations

Example Question #3 : Function & Equivalence Relations

All Theory of Positive Integers Resources

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