Consider the sets s12345 and astimes s given...

Name: consider the sets s12345 and astimes s given bya111213323541455152 a calculate the number of sets bstimes s such that ab and b is a reflexive relation212 b find the number of sets bstimes s 75332
Uploaded: 2022-04-22T13:54:27-08:00
Duration: 10 min 42 s
Channel: Shaiju T
Description: consider the sets s12345 and astimes s given bya111213323541455152 a calculate the number of sets bstimes s such that ab and b is a reflexive relation212 b find the number of sets bstimes s 75332

Consider the sets S={1,2,3,4,5} and AS\times S given byA={(1,1),(1,2),(1,3),(3,2),(3,5),(4,1),(4,5),(5,1),(5,2)} . (a) Calculate the number of sets BS\times S such that AB and B is a reflexive relation.2^12 (b) Find the number of sets BS\times S such that AB and B is a symmetric relation. (c) Determine the number of sets BS\times S such that AB and B is an antisymmetric relation. (d) Suppose BS\times S is a transitive relation such that AB . What is the smallest possible cardinality of B ?

Transcript

00:01 Hello students, let's discuss the question.

00:03 Here a cross a is the cartesian product of a and itself.

00:25 This can be defined as the set of all ordered pairs.

00:38 This is written as a comma a pair.

00:44 A belongs to e this is the answer of this part now size of a cross a in term of n and here a is a set of n elements so size of a cross a is equal to n multiplied so size of a cross a is equal to n multiplied by n which is nothing n square this is the answer of the second part now we have to define a relation r on a so this is relation r on set a is a subset of a that is a subset of a cross a that is a subset of a cross a this is the subset of a cross a this is the answer of the third part.

02:07 Now reflexive relation is defined as ordered pair a comma a belongs to some relation are where a belongs to a set a.

02:33 This is relation and this is set.

02:41 This is the answer of the fourth part.

02:45 Now not reflexive relation is defined as where a belongs to a, there exists at least there exists at least one element.

03:28 A belongs to a such that a comma a does not belong to relation r.

03:40 This is the definition of not reflexive.

03:44 Next we have to recognize by how whether a relation is reflexive from its matrix.

03:54 So a relation are is reflexive if the matrix diagonal elements are one.

04:21 This is the answer of next part.

04:28 Now next, a relation r on set a to be symmetric means it is a reflexive because symmetric means a comma b belongs to r for all a comma b belongs to a.

05:13 Now a relation of set a to be symmetric means if a comma b belongs to r, then b comma a also belongs to r for all a comma b belongs to a...

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