Answer:
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If R is a relation that is transitive and symmetric, then R is reflexive on dom(R)={a∣(∃b)aRb}: if a∈dom(R), then there is b such that aRb, thus bRa by symmetry, so aRa by transitivity.
Note that if R is symmetric, then dom(R)=range(R)={b∣(∃a)aRb}.
Hence, to get an example of a relation R on a set A that is transitive and symmetric but not reflexive (on A), there has to be some a∈A which is not R-related to any b∈A. There are many examples of this:
A={0,1} and R={(0,0)},
not reflexive on A because (1,1)∉R,
A={0,1,2} and R={(0,0),(0,1),(1,0),(1,1)},
not reflexive on A because (2,2)∉R.
Step-by-step explanation:
