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Homework Statement
Suppose ##R## and ##S## are relations on a set ##A##.
If ##R## and ##S## are transitive, is ##R \cup S## transitive? Why?
Homework Equations
The Attempt at a Solution
Suppose that ##a## is an arbitrarily but particularly picked element of ##R \cup S##, then
$$a \in R \ \text{or} \ a \in S$$
By division into cases, suppose ##a \in R##,
Suppose ##b,c## are elements of ##R##, such that ##(a, b) \in R## and ##(b,c) \in R##
By definition of ##R##, ##(a,c) \in R##
I then carried out the exact same steps for the case ##a \in S## and concluded that ##R \cup S## is transitive.
However, I also managed to find a counter example by defining ##A = \{ a,b,c \}##, ##R = \{ (a,b) \}##, ##S = \{ (b,c) \}##, ##R \cup S = \{ (a,b) , (b,c) \}##. The fact that ##b \in R \cap S## suggests to me that my division into cases wasn't done right, but aren't my steps analogous to the division into cases seen in problems with statement variables?
Thanks in advance for any assistance!