You must show that this relation issbrajagopal2690 said:Consider the set of positive rational numbers Q+ . Consider the relation r defined by (x,y) ∈ r<=> x/y ∈ Z. Show that r is a partial order and determine all numbers greater than 1/2.
Suppose that $p/q$ is greater than $1/2$ in this ordering (where $p/q$ is a fraction in its reduced form, so that $p$ and $q$ have no common factors other than $1$). Then $\left.\frac12\middle/\frac pq\right.$ is an integer. Simplify that compound fraction and see what that tells you about $p$ and $q$.sbrajagopal2690 said:... determine all numbers greater than 1/2.
A relation is a set of ordered pairs where the first element in each pair is related to the second element. A function is a special type of relation where each input has exactly one output. In other words, a function is a relation where each input has a unique output.
To determine if a relation is a function, you can perform the vertical line test. This means that if you draw a vertical line anywhere on the graph of the relation and it only intersects the graph at one point, then the relation is a function. Alternatively, you can also check if each input has a unique output in the relation.
The domain of a function is the set of all possible inputs or x-values. The range of a function is the set of all possible outputs or y-values. In other words, the domain is the set of all values that can be plugged into a function, and the range is the set of all values that can be obtained as a result.
To find the inverse of a function, you can switch the x and y values of the original function and then solve for y. The resulting equation will be the inverse function. You can also graph the original function and its inverse on the same coordinate plane, and the inverse function will appear as a reflection of the original function over the line y = x.
A one-to-one function is a function where each input has exactly one unique output. There are no two different inputs that result in the same output. An onto function, also known as a surjective function, is a function where every element in the range is mapped to by at least one element in the domain. In other words, the range of an onto function is equal to its codomain.