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The substitution property of equality, or Leibniz's Law (though the latter term is usually reserved for philosophical contexts), generally states that, if two things are equal, then any property of one, must be a property of the other
If a theory has a predicate that satisfies the Law of Identity and Substitution property, it is common to say that it "has equality," or is "a theory with equality." [ 27 ] The use of "equality" here is a misnomer in that an arbitrary binary predicate that satisfies those properties may not be true equality, and there is no property or list of ...
The converse of this axiom follows from the substitution property of equality. 2) Axiom Schema of Specification (or Separation or Restricted Comprehension ): If z is a set and ϕ {\displaystyle \phi } is any property which may be satisfied by all, some, or no elements of z , then there exists a subset y of z containing just those elements x in ...
The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The signature has a single predicate symbol, usually denoted ∈ {\displaystyle \in } , which is a predicate symbol of arity 2 (a binary relation symbol).
An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as ( a + b ) = ( b + a ) {\displaystyle (a+b)=(b+a)} .
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.