Is x is a wife of y transitive relation??? where x and y belong to the set human beings
I suggest that you read Transitive relation - Wikipedia, the free encyclopedia
A transitive relationship is as follows
$ a > b\ and\ b > c \implies a > c.$
So if X is the wife of Y and X is the wife of Z (a situation against the law in many jurisdictions), does that mean Y is necessarily the wife of Z?
This is a more interesting question that I first thought! A relation "xRy" is transitive if and only if "if both aRb and bRc, then aRc". Suppose Alice is the wife of Robin and Robin is the wife of John. Does it follow that "Alice is the wife of John"?
I intentionally use the name "Robin" because it is fairly gender neutral. If we assume that the wife must be a woman and then must be the wife of a man, or if we assert that one person cannot have more than one wife, the two statements, "Alice is the wife of Robin" and "Robin is the wife of John" cannot both be true. And therefore the statement if aRb and bRc is false and, if the hypothesis of an implication is false the implication itself is true whether of not the conclusion is, this is a transitive relation.
I am not disagreeing. But I went down the path you were going. Let's assume that marriage is not a creature of law or custom, but rather a state of emotional commitment. So we might have the situation that Alice and Beth are emotionally married, as are Beth and Carol. That would say nothing about the emotional state between Alice and Carol, who well might loathe each other. So I believe that, under any common definition of marriage, it is not a transitive relationship. Transitory, perhaps, but transitive, no.
I agree with most of what you are saying except that I would not agree that what you are giving is a "common definition of marriage" and I certainly would not agree that what you are saying is true for any common definition of marriage.
Transitivity is a logical structure that extends equivalence: If we know that the statement If A then B is a true statement, and we also know that If B then C is true, then we may also conclude that the statement If A then C is a true statement. This conclusion is valid because it is based upon a logical law, often referred to in mathematics as The Transitive Law. The law is often stated using the concept of equality rather than simply conditional statements: If A=B and B=C then A=C.
A transitive structure contains a compound hypothesis with two logical components: A=B and B=C. If both component hypotheses are true and the entire conditional statement is valid and true then the conclusion is also true. If either component is false, then the statement is true by vacuous hypothesis just as in a simple conditional statement. This law holds true in logic just as it holds true in mathematics (being a pure application of Logic). from Fitly Spoken