Question

Answer these questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of {2, 9}. f ) Find the least upper bound of {2, 9}, if it exists. g) Find all lower bounds of {60, 72}.

Answer 1

Answer:

(a) maximal elements =27,48,60,72

(b) minimal elements =2,9

(c) greatest element =2,9

(d) least element = Does not exist

(e) upper bounds =18,36,72

(f) least upper bound =72

(g) lower bounds =2,4,6,12

(h) greatest lower bound =12

Answer 2

a) The maximal elements are 27, 48, 60, and 72.

b) The minimal elements are 2 and 9.

c) There exists no greatest element.

d) There exists no least element.

e) The upper bounds of {2, 9} are 18, 36, and 72.

f) The least bound of {2, 9} is 18.

g) The lower bounds of {60, 72} are 2, 4, 6,  and 12.

A poset is a partially ordered set that helps to establish and expand the logical idea of an ordering and arrangement of elements of a set.

Let's consider a poset condition where:

• (X ≤ A) is a poset and A ⊆ X (i.e. A is a subset of X)

Then; we can infer that

• An element a of x will be the maximal element provided that there exists no b ∈ x such that a < b.
• An element a of x will be the minimal element provided that there exists no b ∈ x such that b < a.
• An element a of x will be the greatest element provided that b ≤ a for all b ∈ x.

• An element a of x will be the least element provided that a ≤ b for all b ∈ x.

From the given information in the question, we design a Hasse diagram for answering the question for the poset which can be seen in the image below.

∴

• a) The maximal elements are 27, 48, 60, and 72.
• b) The minimal elements are 2 and 9.
• c) There exists no greatest element.
• d) There exists no least element.
• e) The upper bounds of {2, 9} are 18, 36, and 72.
• f) The least bound of {2, 9} is 18.
• g) The lower bounds of {60, 72} are 2, 4, 6,  and 12.

Learn more about ordered set here:

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