no, let A = {1,2,3}. we say a relation R on A is transitive when forall, if and then

you must check EVERY such pair in the relation and show that the implication holds. (note that an implication is true if the first statement is false, the empty set is transitive for instance, because it fulfils the conditions of transitivity vacuously)

like i said, check every such pair where you have and , (and note that a can equal c, and you can also have a = b = c)R2 and R4 is transitive in the book but I do not see why.

no it is not. here's one pair that doesn't work:Isn't R1 transitive? the book says no.

(1,3) = A,B (3,1) = B,C then (1,1) = A,C?? Please explain. Thanks in advance.

we have (3,1) and (1,3) but there is no (3,3)