Q: Let S = {x, y, z } and R is a relation defined on S such that?

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In summary, a relation in mathematics is a set of ordered pairs that relates elements from one set to elements of another set. It can be represented in set notation as a set of ordered pairs, and is defined by specifying the set of ordered pairs that make up the relation. A function is a special type of relation where each element in the domain is paired with only one element in the range, while a relation may have multiple outputs for a single input. A relation can be reflexive, symmetric, and transitive at the same time, which is known as an equivalence relation.
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Q: Let S = {x, y, z } and R is a relation defined on S such that
R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}
Show that R is reflexive and symmetric as well.
 
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A: R is reflexive because for each element x in set S, (x,x) is in relation R. R is symmetric because for each ordered pair (x,y) in relation R, its inverse (y,x) is also in relation R.
 

FAQ: Q: Let S = {x, y, z } and R is a relation defined on S such that?

1. What is the definition of a relation in mathematics?

A relation in mathematics is a set of ordered pairs that relates elements from one set, called the domain, to elements of another set, called the range.

2. How is a relation represented in set notation?

A relation can be represented in set notation as a set of ordered pairs, where the first element in each pair comes from the domain and the second element comes from the range. For example, if R is a relation from set A to set B, it can be written as R = {(a,b) | a∈A and b∈B}.

3. What is the difference between a function and a relation?

A function is a special type of relation where each element in the domain is paired with only one element in the range. In other words, each input has a unique output. A relation, on the other hand, does not have this restriction and may have multiple outputs for a single input.

4. How is a relation defined on a set?

A relation on a set is defined by specifying the set of ordered pairs that make up the relation. This can be done using set notation or by listing out the ordered pairs.

5. Can a relation be reflexive, symmetric, and transitive at the same time?

Yes, a relation can be reflexive, symmetric, and transitive at the same time. This type of relation is called an equivalence relation and is often used in mathematics to define properties of equality.

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