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2. Rule of replacement - Wikipedia

   en.wikipedia.org/wiki/Rule_of_replacement

   In logic, a rule of replacement[1][2][3] is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a ...

3. Logical equivalence - Wikipedia

   en.wikipedia.org/wiki/Logical_equivalence

   Logical equivalence. In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of and is sometimes expressed as , , , or , depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation ...

4. Axiom of choice - Wikipedia

   en.wikipedia.org/wiki/Axiom_of_choice

   Statement. A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f (A) is an element of A. With this concept, the axiom can be stated: Axiom — For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X ...

5. Equivalents of the Axiom of Choice - Wikipedia

   en.wikipedia.org/wiki/Equivalents_of_the_Axiom...

   The first edition of the book includes over 150 statements in mathematics that are equivalent to the axiom of choice, including some that are novel to the book. [ 1 ] [ 2 ] This edition is divided into two parts, the first involving notions expressed using sets and the second involving classes instead of sets.

6. Element (mathematics) - Wikipedia

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

   Element (mathematics) In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four positive integers ( ), one could say that "3 is an element of A ", expressed notationally as 123.

7. Expression (mathematics) - Wikipedia

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

   In mathematics, an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers (constants), variables, operations, and functions. [1] Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of ...

8. Equality (mathematics) - Wikipedia

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

   In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. [1] Equality between A and B is written A = B, and pronounced " A equals B ". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS).

9. Equating coefficients - Wikipedia

   en.wikipedia.org/wiki/Equating_coefficients

   Equating coefficients. In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term.