por What is an ordered pair?
An ordered pair or double is a set of two elements that are written according to the order established by a certain criterion. This criterion specifies which of the two elements comes first and which comes after.
The ordered pair is denoted as (x,y), where “x” is the first element of the pair and “y” is the second, also called components. In general (x, y) is not the same ordered pair (y, x). And besides order, another important feature of ordered pairs is equality: two ordered pairs (a,b) and (c,d) are equal only if a = c and b = d.
Examples of ordered pairs would be those that are made up of the age and weight of a class of mathematics students. The ordered pair (15, 62) corresponds to a 15-year-old student weighing 62 kilos, different from the improbable pair (62,15).
The concept of an ordered pair is very important in various areas of mathematics, such as the Cartesian plane, fractions, vectors in the plane, relations, and functions. An important aspect is that its elements do not necessarily have to be numeric, for example you can make ordered pairs with:
Country-capital city Name-surname Wife-husband
And many other combinations.
Examples of ordered pairs
fractions
A fraction is represented as the quotient of two integers p/q, for example the fraction ½, which is equivalent to the decimal number 0.5.
However, this fraction is not the only one that represents the decimal 0.5, so do the following:
2/4; 3/6; (–2)/(–4); 20/40; (–1)/(–2)…
In this way, any fraction can be represented as an ordered pair (p,q), where p and q are integers, with p occupying the position of the numerator and q that of the denominator. There is an important restriction and that is that q (the denominator) has to be different from 0, since fractions of the form p/0 are not defined.
And another important condition is that two fractions a/b and c/d are equal as long as:
a∙d=b∙c
Functions and their graphs
A function can be expressed as a set of ordered pairs. For example, when graphing a function on the Cartesian plane, the first element is assigned the position of the independent variable, while the second element is assigned the dependent variable. This is a neat pair.
For the function y = f(x), we can express the ordered pair as [x, f(x)]. For example, consider the starting set:
A= {1, 2, 3, 4}
In this set are the first components of a pair ordered according to the function y = x2. The set of second components is:
B={1, 4, 9, 16}
And the ordered pairs that are formed are:
{(1,1); (2,4); (3, 9); (4;16)}
On the Cartesian coordinate system, the ordered pairs represent the points that are joined by a continuous line to generate the graph of the function.
Vectors in the plane
Vectors can be represented in the Cartesian plane by ordered pairs, where the first element represents the horizontal component “x” and the second the vertical component “y”. To distinguish vectors from points in the plane, they are denoted by bold letters and square brackets are used instead of parentheses, like this:
v =
For example, the vector v = < 4, 7 > has a horizontal component equal to 4 and a vertical component equal to 7. Its graph is:
Note that this vector has its origin coinciding with the origin of the coordinate system (0,0). If the vector has its origin at any other point, it can also be expressed in the form of an ordered pair through a subtraction of ordered pairs, for this see the following sections.
Trades with ordered pairs
Addition
Let be the ordered pairs (a,b) and (c,d). A new ordered pair is obtained by their sum according to:
(a,b) + (c,d) = (a+c,b+d)
neutral element
The neutral element of the addition of ordered pairs is the pair (0,0), since when it is added to the ordered pair (a,b), the sum is the latter:
(a,b) + (0,0) = (a,b)
sum of the opposite
Adding an ordered pair (a,b) with its opposite (-a,-b) gives the ordered pair (0,0):
(a,b) + (–a,–b) = (0,0)
commutativity
The order of the addends does not change the sum:
(a,b) + (c,d) = (c,d) + (a,b)
associativity
The result of adding three ordered pairs is not altered when they are grouped together to perform the operation:
[(a,b) + (c,d)] + (e,f) = (a,b) + [(c,d) + (e, f)]
Ordered Pair Subtraction
Let the ordered pairs be (a,b) and (c,d), the subtraction is carried out as follows:
(a,b) – (c,d) = (a–c,b–d)
Product
In the product there are two options: i) multiply an ordered pair by a constant and ii) multiply two (or more) ordered pairs.
Multiply by a constant
Let k be a constant and the ordered pair (a,b), the product between the constant and the pair is:
k∙(a,b) = (k∙a, k∙b)
Ordered Pair Multiplication
The product between the ordered pairs (a,b) and (c,d) is done as follows:
(a,b) x (c,d) = (ac – bd, bc+ad)
neutral element
The neutral element of the multiplication is (1,0), since when multiplying any pair ordered by it, following the multiplication rule described above, the original pair results:
(a,b) x (1,0)= (a – 0, b + 0) = (a,b)
associativity
Since the order of the factors does not change the product, you can group in different ways to multiply three or more ordered pairs and the result is the same:
[(a,b) x (c,d)] x(e,f) = (a,b)x [(c,d) x (e,f)]
solved exercises
Exercise 1
We have the ordered pairs (x2, x–2) = (16, 2). which is the value of x?
Solution
Applying the equality of ordered pairs, we first obtain:
x2 = 16 ⇒ x1 = 4, x2 = –4
To know which of the two values to choose, use is made of:
x–2 = 2
x = 2 + 2 = 4
Therefore, the required value of x is 4.
Exercise 2
Express as an ordered pair the vector that goes from the point (1, 3) to the point (7, 11) and represent it graphically.
Solution
Be v the searched vector. To determine the ordered pair that represents it, and that contains its coordinates, the coordinates of the point of arrival and the point of origin are subtracted, in that order. So:
v = <7-1; 11-3>=<6; 8>
The vector is represented below v as the one that goes from (1,3) to (7, 11) and the equipollent vector v whose origin is set to the origin of the coordinate system (0,0). As you can see, they have the same direction and meaning.
References
DeepAl. Ordered Pair. Retrieved from: deepai.org. matermobile. Cartesian representation of a vector by means of an ordered pair. Recovered from: matemovil.com. Varsity Tutors. Ordered Pair. Recovered from: varsitytutors.com Sacerdoti, Juan. Relations and functions. Faculty of Engineering. Department of Mathematics. Buenos Aires’ University. Retrieved from: materias.fi.uba.ar. University of Denver. Relations. Retrieved from: math.ucdenver.edu.