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In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...
1. Denotes addition and is read as plus; for example, 3 + 2. 2. Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2. 3.
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x ...
The Mostowski collapse lemma states that for every such R there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (X, R), and the isomorphism is unique. The isomorphism maps each element x of X to the set of images of elements y of X such that y R x (Jech 2003:69).
If f(x)=y, then g(y)=x. The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. [21] An elementary proof runs as follows: If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y ...
[note 1] A subset X of S is said to be closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one may also say that X has the closure property. The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set.
The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zero indicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn (with replacement) from the set {rock, scissors, paper}: If x defeats y, and y defeats z, then x does not ...