Section 1: Propositional Logic
... of the equivalent statement forms ∼q ⇒ ∼p or p ⇒ q, whichever is most convenient for the discussion at hand. • if and only if: The biconditional, p ⇔ q is sometimes stated as “p if and only if q” and written “p iff q”. • sufficient: The expression, “p is sufficient for q”(or “p is a sufficient condi ...
... of the equivalent statement forms ∼q ⇒ ∼p or p ⇒ q, whichever is most convenient for the discussion at hand. • if and only if: The biconditional, p ⇔ q is sometimes stated as “p if and only if q” and written “p iff q”. • sufficient: The expression, “p is sufficient for q”(or “p is a sufficient condi ...
Modular Arithmetic continued
... of letters and punctuation, and we will be using power functions in modular arithmetic with very large moduli. In this set of notes we’re focusing on addition an multiplication, and encryption functions, like the example above, which use these two operations. Our example moduli will be small numbers ...
... of letters and punctuation, and we will be using power functions in modular arithmetic with very large moduli. In this set of notes we’re focusing on addition an multiplication, and encryption functions, like the example above, which use these two operations. Our example moduli will be small numbers ...
preprint - Open Science Framework
... our activity of making mathematical constructions. Like most other varieties of constructivism but unlike finitism, the mathematician who carries out mathematical constructions in his mind is conceived of in an idealised way. For example, Brouwer accepts the natural numbers as a potentially infinite ...
... our activity of making mathematical constructions. Like most other varieties of constructivism but unlike finitism, the mathematician who carries out mathematical constructions in his mind is conceived of in an idealised way. For example, Brouwer accepts the natural numbers as a potentially infinite ...
Predicate logic
... Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m ...
... Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m ...
