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How to Prove It, Sec. 4.4, #18a

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose R is a partial order on A. B1[itex]\subseteq[/itex]A, B2[itex]\subseteq[/itex]A, and
    [itex]\forall[/itex]x[itex]\in[/itex]B1[itex]\exists[/itex]y[itex]\in[/itex]B2(xRy),
    and [itex]\forall[/itex]x[itex]\in[/itex]B2[itex]\exists[/itex]y[itex]\in[/itex]B1(xRy).

    Prove that [itex]\forall[/itex]x[itex]\in[/itex]A, x is an upperbound of B1 if and only if
    x is an upperbound of B2.


    2. Relevant equations
    R is a partial order on A if R is reflexive, transitive and antisymmetric.

    Definition of an upper bound: Suppose R is a partial order on A, B[itex]\subseteq[/itex]A,
    and a [itex]\in[/itex]A. Then a is called an upper bound for B if [itex]\forall[/itex]x[itex]\in[/itex]B(xRa).



    3. The attempt at a solution
    Is this way off base?
    What if A = {(x,y)[itex]\in[/itex]Z X Zl Z are integers}
    R = {(x,y)[itex]\in[/itex]Z X Z l n[itex]\in[/itex]Z, y=2n, x=2n+1 and x≥y}
    B1 = {1, 3, 5, 6, 8} B2 = {0, 2, 4, 7, 9}
    A= {(1,0), (3,2), (5,4), (7,6), (9,8)}

    →Let x be an arbitrary upper bound of B1. Since B1[itex]\subseteq[/itex]A, then x[itex]\in[/itex]A. [itex]\forall[/itex]x[itex]\in[/itex]B1[itex]\exists[/itex]y[itex]\in[/itex]B2(xRy). Also,
    [itex]\forall[/itex]x[itex]\in[/itex]B2[itex]\exists[/itex]y[itex]\in[/itex]B1(xRy). Since
    R is the relation x ≥ y, then (xRy) means x is an upper bound of B2.

    [itex]\leftarrow[/itex]Let x be an arbitrary upper bound of B2. Similar reasoning (or lack thereof) to show reverse if, then statement.
     
  2. jcsd
  3. Oct 18, 2011 #2
    Proof. let x [itex]\in[/itex]A. Suppose x is an upperbound of B1. That means
    yRx for all y[itex]\in[/itex]B1. Let z[itex]\in[/itex]B2. By assumption this z[itex]\in[/itex]B2,
    [itex]\exists[/itex]w[itex]\in[/itex]B1 such that zRw. Since w[itex]\in[/itex]B1 and yRx for
    all y [itex]\in[/itex]B1, we know wRx. By transitive property, zRw and wRx implies zRx.

    [itex]\leftarrow[/itex]Similarly, suppose x is an upperbound of B2.
     
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