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    Reflexive, symmetric, and transitive relations

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    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.
    If you have any question or suggestion, please let me know. Thank you.

    © BrainMass Inc. brainmass.com October 3, 2022, 12:17 am ad1c9bdddf
    https://brainmass.com/math/basic-algebra/reflexive-symmetric-transitive-relations-174028

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    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.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com October 3, 2022, 12:17 am ad1c9bdddf>
    https://brainmass.com/math/basic-algebra/reflexive-symmetric-transitive-relations-174028

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