the answer to the above is 15, but I can only get 12. Say the set A = {1,2,3,4}
I can only get {(1,1), ..., (1,1,1), ..., (1,1,1,1)} which gives me 12.
Where is my reasoning faulty?
Thank you
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Are you talking about binary relations? What does ${(1,1)\dots(1,1,1)\dots(1,1,1,1)}$ mean? – saulspatz Mar 25 '20 at 17:21
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Sorry. Yes, I'm speaking about binary relations. That means the amount of binary relations that I found. To expand them: 12 relations: {(1,1), (2,2), (3,3), (4,4), (1,1,1), (2,2,2), ..., (4,4,4,4)} – toyoda Mar 25 '20 at 17:23
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A binary relation is a set of ordered pairs. Triples like $(1,1,1)$ play no part. Furthermore, you only list one set. You need $15$ sets, although I think the correct answer should be $16$. – saulspatz Mar 25 '20 at 17:25
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What's the reasoning behind 15 sets? – toyoda Mar 25 '20 at 17:31
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You are thinking along the right lines, but you're getting confused. You are correct in that if the relation is both symmetric and anti-symmetric, the only pairs in the relation must be of the form $(a,a)$ with $a\in\{1,2,3,4\}$ However, triples and quadruples play no part; a binary relation is a set of ordered pairs. So, the questions is how many different sets are there whose only elements are of the required form?
Here are some examples: $$\{(1,1)\}\\\{(1,1),(2,2)\}$$
Can you find the others? Although your source says $15$, the right answer is $16$.
saulspatz
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Well: Sets of 1 element (like {(1,1)}): 4 Sets of 2 elements I'd say 6, of 3 elements I'd say 4 and of 4 elements I'd say 1. The 16th can't be {}, can it? Thank you – toyoda Mar 25 '20 at 17:42
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@toyoda Yes, it can be the empty set, why not? $\emptyset$ satisfies (vacuously) both the definitions of symmetry and antisymmetry. – saulspatz Mar 25 '20 at 17:44
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If this answer solved your problem, you should accept it by clicking on the checkmark. That way, others see it has been disposed of. – saulspatz Mar 25 '20 at 17:48
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