Is the relation reflexive, symmetric, transitive

In summary, a relation is reflexive if every element in the domain is related to itself, symmetric if for every pair of elements (a, b), if a is related to b, then b is related to a, and transitive if for every triple of elements (a, b, c), if a is related to b and b is related to c, then a is also related to c. These properties are commonly used to analyze and understand relationships in fields such as mathematics, computer science, and social sciences. Reflexive relations have 1's along the diagonal of the matrix representation, while symmetric relations have 1's both above and below the diagonal.
  • #1
iHeartof12
24
0
Indicate which of the following relations on the given sets are reflexive on a given set, which are symmetric and which are transitive.

{(x,y)[itex]\in[/itex][itex]ZxZ[/itex]: x+y=10}

Tell me if I'm thinking about this correctly

It is not reflexive because the only 5R5.
It is symmetric because any xRy and yRx where x+y=10.
It is not transitive because any xRy and yRx, x is not related to x.
 
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  • #2
all correct
 
  • #3
thank you
 

1. Is the relation reflexive?

A relation is reflexive if every element in the domain is related to itself. To determine if a relation is reflexive, we can check if the diagonal elements of the matrix representation of the relation are all 1's. If they are, then the relation is reflexive.

2. Is the relation symmetric?

A relation is symmetric if for every pair of elements (a, b), if a is related to b, then b is related to a. To determine if a relation is symmetric, we can check if the matrix representation is symmetric along the diagonal. If it is, then the relation is symmetric.

3. Is the relation transitive?

A relation is transitive if for every triple of elements (a, b, c), if a is related to b and b is related to c, then a is also related to c. To determine if a relation is transitive, we can check if for every pair of 1's in the matrix representation, the corresponding element in the transitive closure matrix is also a 1. If this is the case, then the relation is transitive.

4. What is the difference between a reflexive and symmetric relation?

A reflexive relation is one where every element in the domain is related to itself, while a symmetric relation is one where for every pair of elements (a, b), if a is related to b, then b is also related to a. In other words, reflexive relations have 1's along the diagonal of the matrix representation, while symmetric relations have 1's both above and below the diagonal.

5. How can I use the properties of reflexivity, symmetry, and transitivity in real-world situations?

These properties can be used to analyze and understand relationships between objects or concepts in various fields such as mathematics, computer science, and social sciences. For example, in mathematics, these properties can be applied to study the relationships between numbers, sets, and functions. In computer science, they can be used to analyze the efficiency of algorithms and data structures. In social sciences, they can be used to understand the dynamics of human relationships and interactions.

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