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2. Transitive relation - Wikipedia

   en.wikipedia.org/wiki/Transitive_relation

   Symbolic statement. 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 ...

3. Relation (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Relation_(mathematics)

   A transitive relation is irreflexive if and only if it is asymmetric. [12] For example, "is ancestor of" is a transitive relation, while "is parent of" is not. Connected for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, on the natural numbers, < is connected, while "is a divisor of " is not (e.g. neither 5R7 nor 7R5). Strongly connected

4. Equivalence relation - Wikipedia

   en.wikipedia.org/wiki/Equivalence_relation

   In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).

5. Equality (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Equality_(mathematics)

   In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. Equality between A and B is written A = B, and pronounced " A equals B ". In this equality, A and B are the members ...

6. Commutative property - Wikipedia

   en.wikipedia.org/wiki/Commutative_property

   In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more ...

7. Inequality (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Inequality_(mathematics)

   In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [ 1] It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than .

8. Group action - Wikipedia

   en.wikipedia.org/wiki/Group_action

   Group action. The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices. In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that ...

9. Transitive set - Wikipedia

   en.wikipedia.org/wiki/Transitive_set

   Transitive set. In set theory, a branch of mathematics, a set is called transitive if either of the following equivalent conditions hold: whenever , and , then . whenever , and is not an urelement, then is a subset of . Similarly, a class is transitive if every element of is a subset of .