Ordered set (1,2)<<(3,4). But how to write if 1st set contains a single object only?

May 2016
3
2
Wageningen, The Netherlands
To denote the relation between sets, I can write (1,2)<<(3,4) to say that each object in the first set is smaller than each object in the second set. Or does it only mean that 1<3 and 2<4?

Now my questions:
  1. How do I write the relation between set {1} and set (3,4)?
  2. How do I write the relation between (1,3) and (3,4)?
  3. Does the above notation only hold for ordered sets, or also for unordered sets like {a,b}<<{c,d}, where a>b or a<b and c>d or c<d?
For question 1 I belief the answer is {1}<(3,4). For question 2 I belief the answer is (1,3)<(3,4). For question 3 I belief the answer is that the notation also holds for unordered sets.
Looking forward to hearing your answers.

Best,
Thomas
 
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May 2016
3
2
Wageningen, The Netherlands
I should maybe add to question 3 that I am aware that I can write the relation between {a,b} and {c,d} as min{a,b}<min{c,d}. However, I'd like to know if this can also be written as {a,b}<<{c,d}.
 
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mathman

Forum Staff
May 2007
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772
There is universally accepted definition for the kinds of sets you are describing. The author would need to define it precisely.
 
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Country Boy

Math Team
Jan 2015
3,261
899
Alabama
I believe Mathman meant to say "There is NO universally accepted definition for the kinds of sets you are describing."
 
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