# Can a relation be both reflexive and Irreflexive?

## Can a relation be both reflexive and Irreflexive?

Notice that the definitions of reflexive and irreflexive relations are not complementary. That is, a relation on a set may be both reflexive and irreflexive or it may be neither. The same is true for the symmetric and antisymmetric properties, as well as the symmetric and asymmetric properties.

## Can a relation be symmetric and antisymmetric simultaneously?

Some notes on Symmetric and Antisymmetric: • A relation can be both symmetric and antisymmetric. A relation can be neither symmetric nor antisymmetric. Transitive: A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.

Can a relation be reflexive and antisymmetric?

Antisymmetric relations may or may not be reflexive. < is antisymmetric and not reflexive, while the relation “x divides y” is antisymmetric and reflexive, on the set of positive integers. A reflexive relation R on a set A, on the other hand, tells us that we always have (x,x)∈R; everything is related to itself.

How many relations are there on a set with N elements that are reflexive and symmetric?

Now for a Irreflexive relation, (a,a) must not be present in these ordered pairs means total n pairs of (a,a) is not present in R, So number of ordered pairs will be n2-n pairs. So total number of reflexive relations is equal to 2n(n-1). 9. Reflexive and symmetric Relations on a set with n elements : 2n(n-1)/2.

### What is reflexive relation with example?

Reflexive relation on set is a binary element in which every element is related to itself. Consider, for example, a set A = {p, q, r, s}. The relation R1 = {(p, p), (p, r), (q, q), (r, r), (r, s), (s, s)} in A is reflexive, since every element in A is R1-related to itself.

### What is the difference between symmetric and antisymmetric relation?

A binary relation R on a set X is symmetric when : A binary relation R on a set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other; i.e. : ∀a,b∈X((aRb∧bRa)→a=b). An example of antisymmetric relation : The usual order relation ≤ on the real numbers.

What is symmetric relation with example?

A symmetric relation is a type of binary relation. An example is the relation “is equal to”, because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if: If RT represents the converse of R, then R is symmetric if and only if R = RT.

What is antisymmetric relation with example?

Symmetric, Asymmetric and Antisymmetric Relation

Symmetric Asymmetric Antisymmetric
“Is equal to” is a symmetric relation, such as 3 = 2+1 and 1+2=3. “Is less than” is an asymmetric, such as 7<15 but 15 is not less than 7 If a ≠ b, then (b,a)∈R

#### What is meant by Antisymmetric relation?

In mathematics, a homogeneous relation R on set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X.

#### How do you prove Antisymmetric relations?

To prove an antisymmetric relation, we assume that (a, b) and (b, a) are in the relation, and then show that a = b. To prove that our relation, R, is antisymmetric, we assume that a is divisible by b and that b is divisible by a, and we show that a = b.

Is an empty relation symmetric?

Examples. ¨ The empty relation is symmetric, because the statement “if then ” is vacuously true.

Is an empty relation Antisymmetric?

Consequently, if we find distinct elements a and b such that (a,b)∈R and (b,a)∈R, then R is not antisymmetric. The empty relation is the subset ∅. It is clearly irreflexive, hence not reflexive. Likewise, it is antisymmetric and transitive.

## Can a relation be an empty set?

Since there is no such element, it follows that all the elements of the empty set are ordered pairs. Therefore the empty set is a relation. Yes. Every element of the empty set is an ordered pair (vacuously), so the empty set is a set of ordered pairs.

## What is a void relation?

As we know the definition of void relation is that if A be a set, then ϕ ⊆ A×A and so it is a relation on A. This relation is called void relation or empty relation on A. In other words, a relation R on set A is called an empty relation, if no element of A is related to any other element of A.

What are the 3 properties of relation?

Properties of relations

• = is reflexive (2=2)
• = is symmetric (x=2 implies 2=x)
• < is transitive (2<3. and 3<5 implies 2<5)
• < is irreflexive (2<3. implies 2≠3)
• ≤ is antisymmetric (x≤y and y≤x implies x=y)

What is a total relation?

In mathematics, a binary relation R over a set X is total or complete if for all a and b in X, a is related to b or b is related to a (or both).

### What is relation and its properties?

Definition and Properties If the ordered pair of G is reversed, the relation also changes. A binary relation R on a single set A is a subset of A×A. For two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality of a relation R from A to B is mn.

### What is a set relation?

A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x,y) is in the relation. A function is a type of relation.

Is a relation a set?

A relation from a set A to a set B is a subset of A×B. Hence, a relation R consists of ordered pairs (a,b), where a∈A and b∈B….Definition: Relation.

John: MATH 211, CSIT 121, MATH 220
Mary: MATH 230, CSIT 121, MATH 212
Paul: CSIT 120, MATH 230, MATH 220
Sally: MATH 211, CSIT 120

What is difference between identity and reflexive relation?

Thus, in an identity relation, every element is related to itself only. Then R1 is an identity relation on A, but R2 is not an identity relation on A as the element a is related to a and c. Reflexive relation. Every identity relation on a non-empty set A is a reflexive relation, but not conversely.

#### What is the difference between relation and function?

Relation- In maths, the relation is defined as the collection of ordered pairs, which contains an object from one set to the other set. Functions- The relation that defines the set of inputs to the set of outputs is called the functions. In function, each input in the set X has exactly one output in the set Y.

#### How do you tell if a relation is a function?

How To: Given a relationship between two quantities, determine whether the relationship is a function.

1. Identify the input values.
2. Identify the output values.
3. If each input value leads to only one output value, classify the relationship as a function.

Are all function a relation?

The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation which derives one OUTPUT for each given INPUT. Note: All functions are relations, but not all relations are functions.

What is the example of function and relation?

In mathematics, a function can be defined as a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range. For example, y = x + 3 and y = x2 1 are functions because every x-value produces a different y-value. A relation is any set of ordered-pair numbers.

## Which set is a function?

A function is a set of ordered pairs in which no two different ordered pairs have the same x -coordinate. An equation that produces such a set of ordered pairs defines a function.

## What are examples of functions in real life?

You might draw from the following examples:

• A soda, snack, or stamp machine. The user puts in money, punches a specific button, and a specific item drops into the output slot.
• Measurement: Â Thermometer.
• Miles per gallon.
• Basic economics and money math:
• Geometric Patterns.

Is a circle a function?

No. The mathematical formula used to describe a circle is an equation, not one function. For a given set of inputs a function must have at most one output. A circle can be described with two functions, one for the upper half and one for the lower half.

What is the standard form of a circle?

The graph of a circle is completely determined by its center and radius. Standard form for the equation of a circle is (x−h)2+(y−k)2=r2. The center is (h,k) and the radius measures r units. This will result in standard form, from which we can read the circle’s center and radius.

### Are ellipses functions?

An ellipse is not a function because it fails the vertical line test.

### Is a straight line a function?

1 Answer. No, every straight line is not a graph of a function. Nearly all linear equations are functions because they pass the vertical line test. The exceptions are relations that fail the vertical line test.