Angelo G.docx

1. Relations Relation definition. A relation between two sets is a collection of ordered pairs containing one object from each set. If the object is from the first set and the object is from the second set, then the objects are said to be related if the ordered pair is in the relation. A function is a type of relation. Example: For example, {(1, 2), (3, 4), (1, a), (5, r)} is a relation. So is the set {(x, y): y = x2}; this is the set of all ordered pairs (x, y) for which y = x2.

2. Functions A function is a mapping from a set of inputs (the domain) to a set of possible outputs (the codomain). The definition of a functionis based on a set of ordered pairs, where the first element in each pair is from the domain and the second is from the codomain. But, a metaphor that makes the idea of a function easier to understand is the function machine, where an input xx from the domain XX is fed into the machine and the machine spits out the element y=f(x)y=f(x) from the codomain YY. Below, the domain is visualized as a set of spheres and the codomain as a set of cubes, so that the function machine transforms spheres into cubes. Example: Example 2: Number of children A function can output objects of a completely different type than the inputs, as suggested by the above picture where spheres enter the function machine and cubes come out. We could define a function where the domain XX is again the set of people but the codomain is a set of numbers.

For example, let the codomain YY be the set of whole numbers and define the function cc so that for any person xx, the function output c(x)c(x) is the number of children of the person xx. Since there is an upper limit on the number of children a person could possibly have, it's clear the range of cc is not the entire set YY of whole numbers. Putting in the same people into the child number function, we'd obtain c(Martin Luther King, Jr.)=4c(Martin Luther King, Jr.)=4 and c(Marie Skłodowska-Curie)=2. 3. Domain

The domain of a function is the complete set of possible values of the independent variable. In plain English, this definition means: The domain is the set of all possible x-values which will make the function "work", and will output real y-values. When finding the domain, remember:  

The denominator (bottom) of a fraction cannot be zero The number under a square root sign must be positive in this section