Respuesta :
Answer:
(a)R∪T is not Transitive.
(b)R⊕S is irreflexive.
(c)R◦S is Reflexive
Step-by-step explanation:
Definition
Reflexive: A relation on a set A is reflexive if (a,a)∈A for all a∈A.
Transitive: A relation on a set A is said to be transitive if (a, b) ∈A and (b, c)∈A ⇒ (a, c)∈A.
Union: For two sets A and B, A∪B means all elements are either in A or B.
Symmetric Difference A⊕B: These is the set of all elements either in A or in B but not in both.
Composite of A and B (A◦B) consists of all ordered pairs (a,c) such that (a,b)∈B and (b,c)∈A.
(a)R∪T is Transitive
Let a∈R, since R is Reflexive, (a,a)∈R.
By the definition of T, (a, b) ∈T and (b, c)∈T, (a, c)∈T.
R∪T={(a,a),(ac)}
R∪T is not transitive since there is no element b∈R∪T to satisfy the condition for transitivity.
(b)R⊕S is irreflexive
Let a∈R, since R is Reflexive, (a,a)∈R.
Let a∈S, since S is Reflexive, (a,a)∈S.
The symmetric difference contains all elements either in R or in S but not in both.
Therefore: a∉R⊕S.
Therefore R⊕S is irreflexive.
(c)R◦S is reflexive
Let Let a∈R, since R is Reflexive, (a,a)∈R.
Let a∈S, since S is Reflexive, (a,a)∈S.
By the definition of Composite, if
(a,a)∈R and (a,a)∈S, then (a,a)∈R◦S.
Therefore, R◦S is Reflexive since (a,a)∈R◦S for every a∈A.
