Funciones

RELATIONS
Cartesian products are useful for explanation to relate two sets, but when we see mathematics at High School, we pay more attention at relations and functions which are subsets of the Cartesian Product. We use relationships between elements of sets, such as between a student and is or her I.D. number, every day. Relationships are represented using the structure called a relation. 1.Definition A Relation is a set of Ordered Pairs, with a specific condition to choose these ordered pairs (a , b) from a Cartesian Product. Relations are very similar to the predicates already seen in logic. A predicate allows a quality to be attached to a relationship to link two or more objects. A relation is just a relationship between sets of information. Think of all the people in one of yourclasses, and think of their heights. The pairing of names and heights is a relation. In relations, the pairs of names and heights are ordered, which means one comes first and the other comes second. To put it another way, we could set up this pairing so that either you give me a name, and then I give you that person's height, or else you give me a height, and I give you the names of all the peoplewho are that tall. Example: Given A = { Romeo, Adam }and B = { Eve, Juliet }, the Cartesian Product A× B is { (a,b) | a ∈ A, b ∈ B } = {(Romeo, Eve), (Romeo, Juliet), (Adam, Juliet), (Adam, Eve). Now, if we add a condition like R = { (a, b) | a ∈ A, b ∈ B , a loves b}, then R = {(Romeo, Juliet), (Adam, Eve)} A relation R between objects in two sets A and B can be viewed as a subset of the Cartesianproduct A × B, such that R ⊂ A × B. Relations can be represented as sets of ordered pairs, tables, graphs or equations in two variables. 2. Domain and range A relation is described by four components: its domain, its co-domain, its range and a description of the condition to compute the relation. The relation acts as an association between a member of the first set named domain and a member ofthe second set named co-domain.

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The relation from A to B may be represented by R: A → B (a relation from A to B where A is domain, and B is co-domain) Most of the time, only some elements of the co-domain are the result of the mapping, such that the set of all elements representing the complete mapping of the elements from the Domain is called the range. The range is a subset of theco-domain. We can say that the domain is where you start (INPUT); the co-domain is where you go ; the range is where you arrive (OUTPUT). Given A = { Romeo, Adam }and B = { Eve, Juliet, Dalila } and R = { (a, b) | a ∈ A, b ∈ B , a loves b}, then R = {(Romeo, Juliet), (Adam, Eve)} The co-domain is B = { Eve, Juliet, Dalila }, but the range is { Eve, Juliet } If we say “For the domain of natural numbers,and the co-domain of natural numbers, the relation is that the input is the square of the output, we would have a complete description of a relation. We would write this: R: N → N = { (a, b) | a ∈ N , b ∈ N , a = b2 }

This relation is not capable of outputting every member of the co-domain, it can output 1 or 4 or 9 or 16 or any other number which is a perfect square, but not any other naturalnumber. So, although the co-domain is natural numbers, the actual range of outputs is narrower. R = { (1, 2), (2, 4), (3, 9), (4, 16), (5, 25) ……} 3. Mapping diagrams Remember that a map from a set A into a set B is a rule which associates with each element in A a corresponding element in B. Example #1: A = {–2, –1, 1, 2 }, B = { 1, 2, 4 }, R : A → B = { (a, b) | a ∈ A , b ∈ B , b = a2 } R = {(–2, 4), (–1, 1), ( 1, 1), ( 2, 4) } The output b is obtained squaring the input a. Observe that the co-domain is { 1, 2, 4 } and the range is { 1, 4 }. The range is the set of all images. 4 is the image of –2 and the image of 2. 1 is the image of –1 and the image of 1.

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Example #2: A = {–2, –1, 0, 1, 2 }, B = { 1, 4 }, R : B → A = { (b, a) | b ∈ B , a ∈ A , a = ± b } R = { (1, –1), ( 1,...