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Set-builder notation makes use of predicates to define sets. In autoepistemic logic , which rejects the law of excluded middle, predicates may be true, false, or simply unknown . In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a ...
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of ...
For example, type 2 sets have both type 1 members and type 0 members, and extensionality axioms assert that a type 2 set is determined uniquely by either its type 1 members or its type 0 members. Whereas TST has natural models where each type i + 1 {\displaystyle i+1} is the power set of type i {\displaystyle i} , in TTT each type is being ...
This set U is sometimes called the universe of discourse. × (multiplication sign) See also × in § Arithmetic operators. 1. Denotes the Cartesian product of two sets. That is, is the set formed by all pairs of an element of A and an element of B. 2.
[2] [3] Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of ...
The empty set is also occasionally called the null set, [11] though this name is ambiguous and can lead to several interpretations. The power set of a set A, denoted (), is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }.