SHOW WORK! Consider the subset B of R2 given by B = [([0, 1 ] times [0. 3]) [0,
ID: 2985419 • Letter: S
Question
SHOW WORK!
Explanation / Answer
a)
An element g in P is a maximal element if there is no element a in P such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a < m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.
Clearly the minimal element and the minimum of B is (0,0)
The maximal elements are {(1,3),(2,2),(3,1)}
It has no Maximum.
b)
The set [(0,1) x (0,3)] its minima tends to attain (0,0) but since (0,0) is not in the set it cant attain, therefore it doesnot have a least element.So this set has no infimum.
c)Similarly
Take the same set as above [ (0,1) x (0,3) ] .
d)
For a subset A of P, an element x in P is an upper bound of A if a %u2264 x, for each element a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is a lower bound of A if a %u2265 x, for each element a in A. A greatest element of P is an upper bound of P itself, and a least element is a lower bound of P.
e)
For a, b, elements of a partially ordered set P, if a %u2264 b or b %u2264 a, then a and b are comparable. Otherwise they are incomparable.
The required set of points which is incomparable to (1,2) is
B / {[0,1] x [0,2]}.
easy to verify ...
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