Mathematics
Implicit Relations
Implicit relations are equations that define a relationship between variables without explicitly solving for one variable in terms of the others. These equations can be used to represent curves and surfaces in two and three dimensions, respectively. Implicit differentiation is a technique used to find the derivative of an implicitly defined function.
Written by Perlego with AI-assistance
Related key terms
1 of 5
3 Key excerpts on "Implicit Relations"
eBook - PDF- William R. Gondin, Bernard Sohmer(Authors)
- 2014(Publication Date)
- Made Simple(Publisher)
That is the case, for instance, in the equation y = x, where y is expressed as the explicit function, f(x) = Jx. However, the difference between independent and dependent variables is relative. An equation which expresses y as an explicit function of x may be re-written in an equivalent form which expresses x as an explicit function of y. Consider again, for instance, the equation, y=f(x) = ix Interchanging sides and multiplying by 2 = 2, we get the equivalent equation, x = g(y) = 2y This states x as an explicit function of y, and is called the inverse function of y = fix). Equations are often encountered, however, in non-explicit form. They are then said to be implicit equations. Their variables are called co-variables. And the equations are said to express their co-variables as implicit functions of each other. For instance, the equation f(x,y) = x-2y + S = 0 is in implicit form since neither of the co-variables, x or y, stands alone on one side of the equality sign. And the entire expression is to be read: The function, /, of x and y, = x — 2y + 8, is zero'. In a case like this, however, we can derive two equivalent explicit equations from the above implicit equation—one expressing y as an explicit function of x y and the other expressing x as an explicit function of y: y=Â(x) = ix + 4 and x = My) = 2y - 8 It usually shortens the work of computing a table of values if an implicit equation is first changed to explicit form. The following example in a vertical format, for instance, shows arithmetic details of computation in an added central column for the first explicit form of the above implicit equation: X -10 - 1 0 1 10 f(x) = ix + 4 i(-10) + 4 = -5 + 4 K-l) + 4 = «0) + 4 = MD + 4 = -* + 4 0 + 4 + 4 «10) + 4 = 5 + 4 y -1 3* 4 4 9 The dots in this table simply indicate the places where omissions have been made for the sake of brevity.
eBook - PDF- Owen D. Byer, Deirdre L. Smeltzer, Kenneth L. Wantz(Authors)
- 2018(Publication Date)
- American Mathematical Society(Publisher)
5 Relations and Functions Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long as relations do not change. Matter is not important, only form interests them. —Henri Poincaré (1854–1912) It would scarcely be an overstatement to say that most of mathematics, including the contents of this book, can be represented using a mathematical structure called a rela-tion. Nonmathematical relationships—whether familial, business, or romantic—can also be described via relations. They can also describe relationships, whether familial, business, or romantic. Viewed generally as a pairing of elements from two sets, binary relations can describe any personal or mathematical relationship that you can imagine. This includes tasks such as comparing, ordering, grouping, or labeling. Inasmuch as relations describe any assignment of one item to another, functions are merely a special category of rela-tions. Special types of functions, such as permutations and sequences, arise in discrete mathematics and are covered in this text. In this chapter, we provide an introduction to relations, functions, and sequences. 5.1 Relations Let ? and ? be sets, and recall that ?×? is the set of all ordered pairs (?, ?) where ? ∈ ? and ? ∈ ? . A binary relation ? from ? to ? is any subset of ? × ? . We will usually refer to a binary relation as simply a relation . In essence, a relation is any pairing of (some or all of) the elements of one set, ? , with (some or all of) the elements of another set, ? ; symbolically, ? ⊆ ?×? . In the case where ? and ? are equal sets, we simply say that ? is a relation on ? . As above, the letter ? is often used to represent a relation. However, we will also denote relations by mathematical symbols or lowercase letters, which can be Greek (such as ? , ? , or ? ) or Latin (such as ? , ? , or ℎ ). 101 102 Chapter 5 Relations and Functions Example 48. Let ? = {1, 5} and ? = {?, ?, ?} .
eBook - PDF- Carol Whitehead, David A Towers(Authors)
- 2003(Publication Date)
- Red Globe Press(Publisher)
CHAPTER 2 Relations 2.1 Relations on a set Given a set A, we often wish to express the fact that a relation exists between certain pairs of elements of A. The relation is usually expressed in the form of a statement which is true for some pairs of elements in the set and false for the others . Here are some examples. 1. The statement 'a is a factor of b' describes a relation on the set 7L of integers . 2. The statement 'x > y' describes a relation on the set IR of real numbers. 3. The statement 'line I is parallel to line m' describes a relation on the set L of lines in the plane . • There are two special types of relation that are of particular interest in math- ematics and computer science: equivalence relations generalize the idea of equal- ity to 'equal in a certain respect' , and order relations allow us to rank (some) members of a set with respect to a given property. The relation described in Example 2.1.1, 2 is an example of an order relation ; we meet this relation again in Section 5.3. We shall see in Section 2.2 that the relation described in Example 2.1.1, 3 is an example of an equivalence relation. Our lives are so dominated by heirarchical structures that it should be easy to understand why order relations might be important. It may be less obvious at this juncture why equivalence relations should be a key idea in mathematics. We aim to make this clear during the course of this chapter. Notation We denote a relation on a set A by the symbol R (the symbol rv is used by some authors, but we reserve this to denote an equivalence relation which is intro- duced in Section 2.2). Let a, b E A. Then we shall write aRb if 'a is related to b' is a true statement and a P b if 'a is related to b' is false. In Example 2.1.1, 1, A = 7L and R stands for 'is a factor of' . Thus we write aRb to mean 'a is a factor of b'. 32
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
