Then x + x > y + y, that is, twice x is greater than twice y. If x does not equal y, then one of them is greater. Using the same outline that Euclid used to prove proposition I.6. With these axioms, all the properties of magnitudes needed in the first few books of the Elements can be proved. The other half requires the axiomįor each x and y, either x = y, or there is some z such that x + z = y, or there is some z such that x = y + z. If x = y, then x – z = y – z, and w – x = w – y Only a few of these properties are used in Book I. Define a binary relation less than by taking x y means y y. We can now define order in terms of addition. Groups are some of the most important algebraic structures in modern mathematics. When other axioms are added for zero and negation, then the algebra is called a group, and when commutative, an Abelian group. An algebra satisfying only associativity is called a semigroup, while a semigroup that also satisfies commutativity is called a commutative semigroup or an Abelian semigroup. Substitution of equals: If x = y, then x + z = y + z, and z + x = z + y.Īssociativity: For each x, y, and z, ( x + y) + z = x + ( y + z).Ĭommutativity: For each x and y, x + y = y + x.Īssociativity and commutativity together imply that the order that addition is performed is irrelevant. Furthermore, assume addition satisfies the axioms Next, assume a binary operation called addition and written the usual way, x + y. Transitivity: If x = y and y = z, then x = z. Alternatively, we could identify equal magnitudes so that equality is identity.) Assume that equality is what is called an equivalence relation, that is, it satisfies three axioms: (This equality is not identity as we want different magnitudes, such as two different triangles, to be equal. This outline doesn’t have many of the details that would normally be included.įirst, assume there is a binary relation on a set of magnitudes of the same kind called equality, denoted as usual with an equal sign as in x = y. Here is an outline for a presentation for magnitudes. Typically a presentation is given symbolically and in terms of set theory, although the set theory isn’t necessary. In modern mathematics, axioms such as these would form the basis of an abstract algebra. This law, in particular, really ought to have been made an explicit common notion. Is called the law of trichotomy for magnitudes.
The statement thatįor two magnitudes x and y of the same kind, exactly one of the three cases x y holds Statement 2 says that two of those cases cannot simultaneously hold. Some of the others are logical variants of each other, for instance, numbers 1, 8, and 9 are all equivalent to the statement that at least one of the three cases x y holds. Number 11 is a special case of C.N.2 since doubling is a special case of addition, that is, 2 x is just x + x. Number 3 is an instance of the logical principle of double negation, rather than a common notion. If not x = y, then x > y or x y, then x = y or x y. Here are a few of them and locations where they are used. There are a number of properties of magnitudes used in Book I besides the listed Common Notions. At any rate, Euclid frequently treats these two conditions as being equivalent. Symbolically, A > B means that there is some C such that A = B + C. See the notes on I.4 for more discussion on this point.Ĭ.N.5, the whole is greater than the part, could be interpreted as a definition of “greater than.” To say one magnitude B is a part of another A could be taken as saying that A is the sum of B and C for some third magnitude C, the remainder. Using this principle, if one thing can be moved to coincide with another, then they are equal. But the way it traditionally is interpreted is as a justification of a principle of superposition, which is used, for instance, in proposition I.4. On the face of it, it seems to say that if two things are identical (that is, they are the same one), then they are equal, in other words, anything equals itself. For instance, a line cannot be added to a rectangle, nor can an angle be compared to a pentagon.Ĭ.N.4 requires interpretation. Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot. The first Common Notion could be applied to plane figures to say, for instance, that if a triangle equals a rectangle, and the rectangle equals a square, then the triangle also equals the square.
The various kinds of magnitudes that occur in the Elements include lines, angles, plane figures, and solid figures. These common notions, sometimes called axioms, refer to magnitudes of one kind. Things which coincide with one another equal one another. If equals are subtracted from equals, then the remainders are equal.Ĥ. If equals are added to equals, then the wholes are equal.ģ. Things which equal the same thing also equal one another.Ģ.
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