# Reflexive, symmetric, and transitive relations

Indicate which of the following relations on the given sets are reflexive on a given set (see attached)

College level Math Proof before Real Analysis.

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Please see the attached file.

Solution.

(l) The relation T is reflexive, as for every ,

So,

T for every .

(2) The relation is NOT symmetric, as for and ,

T but is not related to , as 1+3>1+2.

(3) The relation is transitive, as for all , if

T and T , then

and

So,

implying that T .

(The material is from Equivalence Relations. Please solve for the part (l) from #1 and every part of # 12 and explain each step of your solutions. Thank you.)

Solution.

(a) We need to prove both

1) If R is a reflexive relation on A, then the identity relation .

Proof: As R is a reflexive relation on A, for every , , which implies that ;

2) If , then R is a reflexive relation on A.

Proof: As , for every , , which implies that R is a reflexive relation on A;

(b) Similar to the proof in a), we need to prove both

1) If R is a symmetric relation on A, then

Proof. As R is a symmetric relation on A, by the definition, for every , if , then ;

Note that if , then So, we proved that for every , This means that . Similarly, we can get Hence,

and

2) If , then R is a symmetric relation on A.

Proof. As , for every , . By the definition, we know that if , then . Hence, we proved that for every , implying that R is a symmetric relation on A.

(c) Similar to the proof in a), we need to prove both

3) If R is a transitive relation on A, then

Proof. As R is a transitive relation on A, by the definition, for every x, y and z in A, if and , then ; That is exactly what meant. So, we proved that if R is a transitive relation on A, then ,

and

4) If , then R is a transitive relation on A.

Proof. As , for every , there is y in A such that and . Hence, we proved that for every x,y,z in A, if , , then . So, R is a transitive relation on A.

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