Domain and Range
... The domain of a relation is the set of first coordinates of the ordered pairs The range of a relation is the set of second coordinates of the ordered pairs We can also visualize this graphically ...
... The domain of a relation is the set of first coordinates of the ordered pairs The range of a relation is the set of second coordinates of the ordered pairs We can also visualize this graphically ...
Relative normalization
... Definition 5 (Pre-model). A pre-model M (of a given signature) is given by – A nonempty set (still) written M and called the domain of M (or several domains Mσ for many-sorted theories); – For each function symbol f of arity n, a function f M : Mn → M; – For each predicate symbol p of arity n, a fun ...
... Definition 5 (Pre-model). A pre-model M (of a given signature) is given by – A nonempty set (still) written M and called the domain of M (or several domains Mσ for many-sorted theories); – For each function symbol f of arity n, a function f M : Mn → M; – For each predicate symbol p of arity n, a fun ...
View Full Course Description - University of Nebraska–Lincoln
... Target Audience: This course was designed for teachers of middle level mathematics. Comfort with functions, algebraic representations and basic logic is strongly recommended. Graduate Credit Hours: 3 Usual Delivery Format: On-site summer course. Designed to be completed in one week, meeting M – F fr ...
... Target Audience: This course was designed for teachers of middle level mathematics. Comfort with functions, algebraic representations and basic logic is strongly recommended. Graduate Credit Hours: 3 Usual Delivery Format: On-site summer course. Designed to be completed in one week, meeting M – F fr ...
Mathematics in Rubik`s cube.
... • G is a set of objects, * is an operator acting on them • 4 axioms: • Closed (for any group elements a and b, a*b ∈ G) • Operation * is associative • For elements a, b, and c, (a*b)*c=a(b*c) ...
... • G is a set of objects, * is an operator acting on them • 4 axioms: • Closed (for any group elements a and b, a*b ∈ G) • Operation * is associative • For elements a, b, and c, (a*b)*c=a(b*c) ...
PDF
... integers of K. OK is a commutative ring with identity. See examples of ring of integers of a number field. 3. Real and complex embeddings of a number field. Read also about totally real and imaginary fields. 4. Norm and trace of an algebraic number. See also this entry. One also can take the norm of ...
... integers of K. OK is a commutative ring with identity. See examples of ring of integers of a number field. 3. Real and complex embeddings of a number field. Read also about totally real and imaginary fields. 4. Norm and trace of an algebraic number. See also this entry. One also can take the norm of ...
on Computability
... Godel's Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the sys ...
... Godel's Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the sys ...
Axiomatic Systems
... 1. Given any two distinct points, there is exactly one line containing both of them. 2. Given any two distinct lines, they intersect in a single point. 3. There exist four points, no three of which are contained in a common line. 4. The total number of points is nite. Exercise 1: Try to come up wit ...
... 1. Given any two distinct points, there is exactly one line containing both of them. 2. Given any two distinct lines, they intersect in a single point. 3. There exist four points, no three of which are contained in a common line. 4. The total number of points is nite. Exercise 1: Try to come up wit ...
Special Facts to Know
... Fibonacci – F0 = 0, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Lucas – F0 = 2, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Let s(n) be the sum of all the proper factors of n. Deficient – s(n) < n Perfect – s(n) = n Abundant – s(n) > n Let d(n) be the total number of digits in the prime factorization of n. Frugal / E ...
... Fibonacci – F0 = 0, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Lucas – F0 = 2, F1 = 1, Fn = Fn-1 + Fn-2 for n > 1 Let s(n) be the sum of all the proper factors of n. Deficient – s(n) < n Perfect – s(n) = n Abundant – s(n) > n Let d(n) be the total number of digits in the prime factorization of n. Frugal / E ...
Lecture 16 Notes
... We will look briefly at the incompleteness result since that has received a good deal of attention as a contrast to the Gödel completeness result for classical FOL. The folklore has it that Gödel’s result cannot be constructive.1 We will not explore Kripke models and the important result of Vim Ve ...
... We will look briefly at the incompleteness result since that has received a good deal of attention as a contrast to the Gödel completeness result for classical FOL. The folklore has it that Gödel’s result cannot be constructive.1 We will not explore Kripke models and the important result of Vim Ve ...
Handout on Revenge
... Consider a sentence pγq, which says: either pγq is unhealthy or untrue. Suppose, for contradiction, that pγq is healthy. Then by the restricted T-schema, SRT, pγq would be true if and only if it were either untrue or unhealthy (T r(pγq) ↔ (¬H(pγq) ∨ ¬T r(pγq)). But this can only happen if pγq is unh ...
... Consider a sentence pγq, which says: either pγq is unhealthy or untrue. Suppose, for contradiction, that pγq is healthy. Then by the restricted T-schema, SRT, pγq would be true if and only if it were either untrue or unhealthy (T r(pγq) ↔ (¬H(pγq) ∨ ¬T r(pγq)). But this can only happen if pγq is unh ...