Equality in Mathematics
This is a guide on equality in mathematics.
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The following document presents and expands upon the mathematical concepts of equality, equivalence, the relationship between mathematical and physical descriptions, and the elementary theory of sets. It is intended to serve as a clear and thorough introduction to these ideas, and each section has been developed in greater detail than in the original so that the reader may follow the reasoning comfortably and see how the various notions connect to one another.
We begin with the notion of equality between objects. In mathematics, when we say that two objects are equal, we mean that they are one and the same object; the two expressions used to describe the object may look different, but they refer to a single entity. This is an important point, because mathematical writing frequently presents the same object in more than one form, and equality is the assertion that these differing forms designate the same thing. For example, when we write that the sum of two and three equals six minus one, we are stating that the number obtained by adding two and three is the very same number as the number obtained by subtracting one from six. Both procedures yield the number five, and the equality records the fact that the results coincide. The expressions on either side of the equals sign are not themselves identical in appearance, but they denote the same number, and that is precisely what the word equality is meant to convey.
The word equivalent is used in several distinct contexts, and it is helpful to distinguish them carefully. In the first and most common context, equivalence applies to assertions, that is, to statements that may be either true or false. When we say that two assertions are equivalent, we mean that the first is true if and only if the second is true; in other words, the two assertions always share the same truth value, so that whenever one holds the other also holds, and whenever one fails the other also fails. For instance, the assertion that a number x satisfies the equation two x plus five equals three is equivalent to the assertion that x is equal to negative one, because the equation is satisfied exactly when x takes that value and by no other value. The word equivalent will appear in other mathematical contexts as well, and in each case the meaning will be explained when the need for it arises, so that no ambiguity is allowed to persist.
The mathematics discussed here has many applications and counterparts in the physical world. Numbers, for example, can be used to measure length, area, speed, and density, among many other quantities. For the sake of clarity, language is used in such a way that mathematical notions are not ordinarily identified with their physical counterparts. Instead, words such as correspond and represent are employed when the intention is to associate a physical quantity with a mathematical one. This careful separation of vocabulary allows mathematical objects to be discussed on two distinct levels: the purely logical level of axioms, deductions, and proofs, and the mixed physical level at which mathematical structures are connected to observable quantities. It is often tedious and not necessarily illuminating to insist that every argument proceed by strictly logical procedures alone. Following physical intuition can be useful and is often more pleasant, provided that the reader remains aware of which level of discourse is being used at any given moment.
In accordance with standard mathematical terminology, a collection of objects is called a set, and the objects belonging to that collection are called the elements of the set. The set of all real numbers is denoted by the symbol R, and the statement that x is an element of R means the same thing as the statement that x is a real number. These conventions provide a compact and precise vocabulary for referring to collections of numbers and to the individual numbers that belong to them.
The notion of a subset extends this vocabulary in a natural way. If S and T are sets, then S is said to be a subset of T if every element of S is also an element of T. Several familiar examples illustrate the idea. The set of rational numbers is a subset of the set of real numbers, since every rational number is a real number. The set of integers is a subset of the set of rational numbers, and it is therefore also a subset of the set of real numbers. In ordinary language, the set of boys is a subset of the set of all children. Likewise, the set of all real numbers x satisfying the inequality two x plus three is less than five is a subset of the real numbers. By convention, a subset of a set S is allowed to be all of S, so that for any set S, the set S is a subset of itself. This convention is convenient because it allows statements about subsets to be made without constantly excluding the case in which the two sets coincide.
A set is often described by stating the conditions under which something is an element of the set. Sometimes these conditions are stated in such a way that no object satisfies them at all. For example, there is no element in the set of all numbers x that satisfy both the condition that x is less than zero and the condition that x is greater than zero, since no number can be simultaneously negative and positive. Similarly, there is no element in the set of all positive numbers x that satisfy both the condition that two x divided by x plus one is greater than one and the condition that x is less than one half. Whenever a set has no elements, it is said to be empty, and this terminology applies in every such case. Thus the set of numbers x such that two x is greater than one and x is less than negative three is empty, because the two conditions cannot be satisfied at once.
Finally, when S and S prime are sets, it is often necessary to prove that they are equal. A standard method for doing so is to prove that S is a subset of S prime and also that S prime is a subset of S; together these two containments establish that the sets have exactly the same elements and are therefore the same set. An example makes the method concrete. Let S be the set of numbers x such that x lies between one and two inclusive, and let T be the set of all numbers of the form five x where x belongs to S. It may be contended that T is the set of numbers y such that y lies between five and ten inclusive. To establish this, let T prime be the set of all numbers y satisfying that condition. Every element of T is obtained by multiplying an element of S by five, and since each such x lies between one and two, the resulting product lies between five and ten, so T is contained in T prime. Conversely, let y be a point of T prime and assume that y lies between five and ten. Setting x equal to y divided by five, it follows that x lies in S and that y equals five x, so T prime is contained in T. The two containments together prove that T equals T prime, and the argument illustrates how the definition of set equality is applied in practice.
The principal points developed above may be summarized as follows:
Equality between objects means that they are the same object, even when they are described by different expressions.
Two assertions are equivalent when each is true if and only if the other is true.
Mathematical notions are deliberately distinguished from their physical counterparts through words such as correspond and represent, and mathematical work may proceed on either a strictly logical level or a mixed physical level.
A set is a collection of objects called elements, and the set of all real numbers is denoted by R.
A set S is a subset of a set T when every element of S is also an element of T, and every set is a subset of itself by convention.
A set may be described by conditions on its elements, and when no object satisfies those conditions the set is called empty.
To prove that two sets are equal, it suffices to prove that each is a subset of the other.