With a> 0, of course.dpa said:Homework Statement
Q. Define an order relation
(x,y)<(j,k) if and only if x+k<y+j
Homework Equations
x<y means x+a=y and viceversa
An order relation only has to be transitive.The Attempt at a Solution
I have no idea. To show it is equivalence relation, I simply show that it is reflexive, transitive and symmetric. but how do I show it is order relation?
An order relation is a mathematical concept that describes the relationship between two elements in a set. It determines the relative positions of the elements in terms of "greater than", "less than", or "equal to" relationships.
An order relation must satisfy three properties: reflexivity, transitivity, and antisymmetry. Reflexivity means that an element is always related to itself. Transitivity means that if element A is related to element B and element B is related to element C, then element A is also related to element C. Antisymmetry means that if element A is related to element B and element B is related to element A, then element A must be equal to element B.
An order relation can be represented using symbols, such as ">", "<", or "=". For example, if element A is greater than element B, it can be represented as A > B. In some cases, order relations can also be represented using diagrams, such as Hasse diagrams.
There are two main types of order relations: total order and partial order. In a total order, all elements in a set are related to each other, and there is a defined "greater than" and "less than" relationship between them. In a partial order, there may be elements that are not related to each other, and the "greater than" and "less than" relationships may not be fully defined.
An order relation has many practical applications in fields such as computer science, economics, and physics. It is used to sort data, classify objects, and analyze relationships between different elements. For example, in computer science, order relations are used to sort algorithms based on their efficiency, and in economics, they are used to rank different options based on their cost and benefits.