Discrete Mathematics, Chapter 5: Induction and Recursion
... Recursively Defined Sets and Structures Recursive definitions of sets have two parts: The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has ...
... Recursively Defined Sets and Structures Recursive definitions of sets have two parts: The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has ...
Autonomous categories and linear logic
... an appendix to that volume. At the time, the formal construction appeared not to have substantial mathematical interest, but it appears to be the most interesting part in the present context. It turns out, for example, that one can in many cases construct models of the full linear logic from Chu’s c ...
... an appendix to that volume. At the time, the formal construction appeared not to have substantial mathematical interest, but it appears to be the most interesting part in the present context. It turns out, for example, that one can in many cases construct models of the full linear logic from Chu’s c ...
The Dedekind Reals in Abstract Stone Duality
... space 1, the space of natural numbers N and the Sierpiński space Σ, which are axiomatised in terms of their universal properties. (Recall that, classically, the Sierpiński space has one open and one closed point.) We can form products of spaces, X × Y , and exponentials of the form ΣX , but not ar ...
... space 1, the space of natural numbers N and the Sierpiński space Σ, which are axiomatised in terms of their universal properties. (Recall that, classically, the Sierpiński space has one open and one closed point.) We can form products of spaces, X × Y , and exponentials of the form ΣX , but not ar ...
Chapter 5 - Stanford Lagunita
... that a sentence Q is a logical truth, then you may assert Q at any point in your proof. We already saw this principle at work in Chapter 2, when we discussed the reflexivity of identity, the principle that allowed us to assert a sentence of the form a = a at any point in a proof. It also allows us t ...
... that a sentence Q is a logical truth, then you may assert Q at any point in your proof. We already saw this principle at work in Chapter 2, when we discussed the reflexivity of identity, the principle that allowed us to assert a sentence of the form a = a at any point in a proof. It also allows us t ...
Composition algebras of degree two
... finite dimensional subspace of A where the bilinear form is nondegenerate and dim B > 1. The previous lemma says that B1 = [x e B | n(x)3 — (x, x2)2 / 0} is nonempty, so it is an open set in the Zariski topology of B. Because any element in B generates a composition subalgebra of A of dimension two, ...
... finite dimensional subspace of A where the bilinear form is nondegenerate and dim B > 1. The previous lemma says that B1 = [x e B | n(x)3 — (x, x2)2 / 0} is nonempty, so it is an open set in the Zariski topology of B. Because any element in B generates a composition subalgebra of A of dimension two, ...
Dependent Types In Lambda Cube
... known from the simply typed lambda calculus. Formula A ∧ B can be seen as a product type, which consists of types A and B, which themselves are again formulas on the side of logic, and types on the side of type theory. ◦ connective and type constructor: We could see already one example above, namely ...
... known from the simply typed lambda calculus. Formula A ∧ B can be seen as a product type, which consists of types A and B, which themselves are again formulas on the side of logic, and types on the side of type theory. ◦ connective and type constructor: We could see already one example above, namely ...
On the Complexity of Linking Deductive and Abstract Argument
... important, and of course by relaxing this constraint we admit into our analysis some scenarios that do not seem to have any useful interpretation; but of course this does not invalidate the results we present. Let A(∆) denote the set of arguments over ∆. If α is an argument, then we denote the suppo ...
... important, and of course by relaxing this constraint we admit into our analysis some scenarios that do not seem to have any useful interpretation; but of course this does not invalidate the results we present. Let A(∆) denote the set of arguments over ∆. If α is an argument, then we denote the suppo ...
MA3A9. Students will use sequences and series
... a. Investigate and explore characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. MA3A4. Students will investigate functions. a. Compare a ...
... a. Investigate and explore characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. MA3A4. Students will investigate functions. a. Compare a ...
Section
... the trap of thinking that distinct variables range over distinct objects. In fact, the sentence xyP(x,y) logically implies xP(x,x), and xP(x,x) logically implies xyP(x,y)! ...
... the trap of thinking that distinct variables range over distinct objects. In fact, the sentence xyP(x,y) logically implies xP(x,x), and xP(x,x) logically implies xyP(x,y)! ...
