basic examples of relations -- proof writing examples 16

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[M.athematech]

second last example its not transitive, c is related to b and b is related to c but c isn't related to c

[Happy_Abe]

@8:49
You can’t claim it’s transitive from one example. bRc and cRb but b is not related to b so it’s not transitive

[tangpiseth]

At 10:45, regarding box 3, I don't understand how R is symmetric and transitive. Although I have a hunch that it is indeed the case and I need help with the following justifications.

R is symmetric because for every element a and b in the set A = {1, 2, 3, 6}, if aRb, then bRa. From R = {(a, a)}, we deduce that aRa. It follows that in this case, b = a, then the statement "if aRb, then bRa" does hold due to the non-existence of any other elements besides "a" that is related to "a". The intuition is that the antecedent, namely, "if aRb (is the case)" isn't even true due to R = {(a, a)}.


R is transitive because for every element a, b and c in the set A, if aRb and bRc, then aRc. From R = {(a, a)}, we observe that aRa. Similar to the above argument, b = a and c = a, then the statement "if aRa and bRc, then aRc" is indeed true since no other elements in the set A are even let the antecedent, "if aRb and bRc", be true. Only case is when, as I had mentioned, b =a and c = a.

END OF PROOFS.

[r.maelstrom]

Box 9 IS transitive. so it is box 10 too.

[M.athematech]

last example, 9, 10, 11, 12 are all transitive

[ScouseRobert]

Just wondering whether the empty set relation qualifies as symmetric as it never violates aRb implies bRa?
Also does it satisfy transitivity for the same reason?