Hi, I'm reading a book on sets and it mentions a set B = {1,2,3,4} and it says that R3 = {(x, y) : x ∈ B ∧y ∈ B} What does that mean? Does that mean every possible combination in the set? Also the book doesn't clarify this completely but for example using the set B say i had another set R = {(1,2),(2,3),(1,3),(1,1),(2,2),(3,3),(4,4)}, Would this be clarified as transitive and reflexive? My question is does a set need to have all transitive properties and all the reflexive properties to be called transitive and reflexive. If i had another set: R1 = {(1,2),(2,3),(1,3),(1,1),(2,2),(3,3)} In which i removed (4,4) would this set R1 still be considered reflexive? Thanks in advance
Your first example is a transitive and reflexive relation. A relation is transitive and reflexive if it satisfies the axioms for transitivity and reflexivity. Your other example is not reflexive, since 4 is an element of X, but 4 ~ 4 is not satisfied.
You appear to have the wrong idea about the "transitive" and "reflexive" properties. You cannot talk about "all the transitive properties" and "all the reflexive properties" because there is only one of each. We apply the term "reflexive" to the whole relation, not individual pairs. If we have a relation on set A, then it is a subset of AxA, the set of all ordered pairs with each member from set A. Such a relation is called "reflexive" if and only if, for every a in A, (a, a) is in the relation. If a particular such pair, say, (x, x), is in the relation, we do NOT call that pair "a reflexive property". Similarly, a relation is called "transitive" if and only if whenever pairs (a, b) and (b, c) are in the relation, so is (a, c). We do NOT apply the term "transitive" to the individual pairs.