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Math Help - Order Relations and Predecessors

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    Order Relations and Predecessors

    Let Z+ denote the set of positive integers. Consider the following order relations on Z+ x Z+:
    (1) the dictionary order
    (2) (x0,y0)<(x1,y1) if either x0-y0<x1-y1, or x0-y0=x1-y1 and y0<y1.
    (3) (x0,y0)<(x1,y1) if either x0+y0<x1+y1, or x0+y0=x1+y1 and y0<y1.

    Questions: in these order relations, which elements have immediate predecessors? Does the set have a smallest element? Show that all three order types are different.
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    Quote Originally Posted by tn11631 View Post
    Let Z+ denote the set of positive integers. Consider the following order relations on Z+ x Z+:
    (1) the dictionary order
    (2) (x0,y0)<(x1,y1) if either x0-y0<x1-y1, or x0-y0=x1-y1 and y0<y1.
    (3) (x0,y0)<(x1,y1) if either x0+y0<x1+y1, or x0+y0=x1+y1 and y0<y1.

    Questions: in these order relations, which elements have immediate predecessors? Does the set have a smallest element? Show that all three order types are different.
    (1)

    All elements have a predecessor except for (x,1) for all positive integers x. Smallest element is (1,1).
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