Respuesta :
Answer:
For the set X = {a, b, c}, the following three relations satisfy the required conditions in (a), (b) and (c) respectively.
(a) R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)} is reflexive and symmetric but not transitive .
(b) R = {(a, a), (b, b), (c, c), (a, b)} is reflexive and transitive but not symmetric .
(c) R = {(a,a), (a, b), (b, a)} is symmetric and transitive but not reflexive .
Step-by-step explanation:
Before, we go on to check these relations for the desired properties, let us define what it means for a relation to be reflexive, symmetric or transitive.
Given a relation R on a set X,
R is said to be reflexive if for every [tex]a \in X, (a,a) \in R[/tex].
R is said to be symmetric if for every [tex](a, b) \in R, (b, a) \in R[/tex].
R is said to be transitive if [tex](a, b) \in R[/tex] and [tex](b, c) \in R[/tex], then [tex](a, c) \in R[/tex].
(a) Let R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)}.
Reflexive: [tex](a, a), (b, b), (c, c) \in R[/tex]
Therefore, R is reflexive.
Symmetric: [tex](a, b) \in R \implies (b, a) \in R[/tex]
Therefore R is symmetric.
Transitive: [tex](a, b) \in R \ and \ (b, c) \in R[/tex] but but (a,c) is not in R.
Therefore, R is not transitive.
Therefore, R is reflexive and symmetric but not transitive .
(b) R = {(a, a), (b, b), (c, c), (a, b)}
Reflexive: [tex](a, a), (b, b) \ and \ (c, c) \in R[/tex]
Therefore, R is reflexive.
Symmetric: [tex](a, b) \in R \ but \ (b, a) \not \in R[/tex]
Therefore R is not symmetric.
Transitive: [tex](a, a), (a, b) \in R[/tex] and [tex](a, b) \in R[/tex].
Therefore, R is transitive.
Therefore, R is reflexive and transitive but not symmetric .
(c) R = {(a,a), (a, b), (b, a)}
Reflexive: [tex](a, a) \in R[/tex] but (b, b) and (c, c) are not in R
R must contain all ordered pairs of the form (x, x) for all x in R to be considered reflexive.
Therefore, R is not reflexive.
Symmetric: [tex](a, b) \in R[/tex] and [tex](b, a) \in R[/tex]
Therefore R is symmetric.
Transitive: [tex](a, a), (a, b) \in R[/tex] and [tex](a, b) \in R[/tex].
Therefore, R is transitive.
Therefore, R is symmetric and transitive but not reflexive .
Relation from the set of two variables is subset of certain product. The relation for the condition are,
[tex]R_1\;\;\;\;\ (1,1), (1,2),,(2,1), (2,2),(2,3)((3,2),3,3)[/tex]
[tex]R_2\;\;\;\;\ (1,1), (2,2),,(3,3)(1,3)3,1)[/tex]
[tex]R_3\;\;\;\;\ (1,2),,(2,1), ,(2,3)((3,2)[/tex]
Relation-
Relation from the set of two variables is subset of certain product. Relation are of three types-
- Reflexive
- Symmetric
- Transitive
1) Reflexive and symmetric but not transitive -
Let a data set as,
[tex]X=1,2,3[/tex]
For the data set the relation can be given as,
[tex]R_1\;\;\;\;\ (1,1), (1,2),,(2,1), (2,2),(2,3)((3,2),3,3)[/tex]
[tex]R_1[/tex] is reflexive as it can be represent as [tex]R_1(a,a)[/tex] for,
[tex]a=1,2,3, \;\;\;\;\; [/tex]
[tex]a[/tex] ∈ [tex]X[/tex]
[tex]R_1[/tex] is symmetric as it can be represent as [tex]R_1(a,b)[/tex] for,
[tex]a,b \;\;\;\;(1,2) (2,1)[/tex]
[tex]a,b[/tex] ∈ [tex]X[/tex]
[tex]R_1[/tex] is not transitive as it can be represent as [tex]R_1\neq (a,c)[/tex] .
[tex]a,c\neq \;\;\;\;(1,3) (3,1)[/tex]
2) Reflexive and transitive but not symmetric
Let a data set as,
[tex]X=1,2,3[/tex]
For the data set the relation can be given as,
[tex]R_2\;\;\;\;\ (1,1), (2,2),,(3,3)(1,3)3,1)[/tex]
[tex]R_2[/tex] is reflexive as it can be represent as [tex]R_2(a,a)[/tex] for,
[tex]a=1,2,3, \;\;\;\;\; [/tex]
[tex]a[/tex] ∈ [tex]X[/tex]
[tex]R_1[/tex] is transitive as it can be represent as [tex]R_1(a,c)[/tex] for,
[tex]a,c \;\;\;\;(1,3) (3,1)[/tex]
[tex]a,c[/tex] ∈ [tex]X[/tex]
[tex]R_1[/tex] is not symmetric as it can be represent as [tex]R_1\neq (a,b)[/tex] .
[tex]a,b\neq \;\;\;\;(1,2) (2,1)[/tex]
3) Symmetric and transitive but not reflexive
Let a data set as,
[tex]X=1,2,3[/tex]
For the data set the relation can be given as,
[tex]R_3\;\;\;\;\ (1,2),,(2,1), ,(2,3)((3,2)[/tex]
[tex]R_1[/tex] is symmetric as it can be represent as [tex]R_3(a,b)[/tex] for,
[tex]a,b=(1,2),(2,1) \;\;\;\;\; [/tex]
[tex]a,b[/tex] ∈ [tex]X[/tex]
[tex]R_3[/tex] is transitive as it can be represent as [tex]R_3(a,c)[/tex] for,
[tex]a,c \;\;\;\;(1,3) (3,1)[/tex]
[tex]a,c[/tex] ∈ [tex]X[/tex]
[tex]R_1[/tex] is not reflexive as it can be represent as [tex]R_3\neq (a,a)[/tex] .
[tex]a,a\neq \;\;\;\;(1,1) [/tex]
Thus the relation for the condition are,
[tex]R_1\;\;\;\;\ (1,1), (1,2),,(2,1), (2,2),(2,3)((3,2),3,3)[/tex]
[tex]R_2\;\;\;\;\ (1,1), (2,2),,(3,3)(1,3)3,1)[/tex]
[tex]R_3\;\;\;\;\ (1,2),,(2,1), ,(2,3)((3,2)[/tex]
Learn more about the Reflexive, symmetric and transitive relation here;
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