Answer to Question #238843 in Discrete Mathematics for lavanya

Question #238843

What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order


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Expert's answer
2021-09-27T12:27:27-0400

P(S)={,{x},{y},{z},{x,y},{x,z},{y,z},{x,y,z}}P(S)=\{\empty,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}\}

Define a relation R by A R B iff AB  A,BP(S)A\sube B \ \forall\ A,B\isin P(S)


We have AAA\sube A for any AP(S)A\isin P(S)

    \implies \sube is reflexive on P(S)P(S)


For any A,BP(S), AB,BAA,B\isin P(S),\ A\sube B,B\sube A then A=BA=B

    R\implies R is anti symmetric on P(S)P(S)


For any A,B,CP(S), AB,BC    ACA,B,C\isin P(S),\ A\sube B,B\sube C\implies A\sube C then A=BA=B

    R\implies R is transitive on P(S)P(S)


So, R()R(\sube) is a partial order on P(S)P(S).


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