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How do I create a R over {a,b,c,d} such that a is maximal but not the greatest element?

I know that something like R{(a,a)(b,b)(c,c)(d,d)}, all the element are maximal.

CJ408
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  • Nope, what you have is an example where every element is maximal but none is greatest. Maximal means 'there is no bigger'. Greatest means 'bigger than every other'. –  Jan 25 '18 at 01:33
  • thanks, i got them mixed up, but i don't know how to create one where one element is the maximal but not the greatest – CJ408 Jan 25 '18 at 01:37
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    What's wrong with the example you gave in your question? – bof Jan 25 '18 at 01:41
  • @CJ408 So you want one element to be maximal but not the greatest, and ... what about the other elements? Are some of them allowed to be maximal (but not greatest) too? If yes - your example is good (you have four elements, all maximal but not greatest). –  Jan 25 '18 at 01:49
  • @CJ408 If not, I cannot help you: if you have only one maximal element, in a finite set, it must be the greatest. (Proof: Start from any element, either it is maximal, or there is bigger, go to bigger etc. - sooner or later you reach the maximal element, which is therefore bigger than your arbitrarily chosen element. Thus, this (unique) maximal element is the greatest.) –  Jan 25 '18 at 01:50

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To clarify, the relation you propose

$$R \equiv \{(a,a), (b,b), (c,c), (d,d)\}$$

has four maximal elements $a,b,c,d$ and no greatest element.

I think you're looking for a relation $R$ where $a$ is a maximal element and no other element is maximal. But on a finite set, if you have a maximal element $a$, and no other element is maximal, then $a$ is the greatest element. You can't have a single maximal element on a finite set without having a greatest element.

My understanding is that your problem is only asking you to find a relation $R$ in which $a$ is a maximal— but not greatest— element, and that it's alright (indeed necessary) if there are other maximal elements as well.

user326210
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