Contrapositive 反證 if φ ⇒ ψ , then ¬ψ ⇒ ¬φ $∃x$A(x)] = $∀x$[¬A(x)] ? Assignment 7 Prove or disprove the statement “All birds can fly.” birds b, fly birds are F(b) Assume $∀b$ ⇒ F(b) True. $∃o$, there are some birds, such as ostrich o, it can not fly. $∀b$[$∃o$$(b=o)$ ⇒ ¬F(b)] so not all birds could fly, $∀b$ ⇒ F(b) False. Prove or disprove the claim (∀x, y ∈ R)[(x − y)2 > 0]. Assume (∀x, y ∈ R)[$(x − y)^2$ > 0] is True. ...
Induction: the inference of a general law from particular instances. Axiom/ principle: generally acknowledged truth. to prove $∀n$ A$(n)$ Method, principle of mathematical induction Prove A$(1)$ Dominoes: $∀n$ [ A$(n)$ ⇒ A$(n+1)$] induction step Theorem: for any n, 1+2+3 …… + n = 1/2n(n+1) if n = 1, both side equal to 1, then the identity is true. Assume A$(n)$ and deduce A$(n+1)$, the identity holds for all n 1+2+3 …… + n+1 = 1/2(n+1)[(n+1)+1] ...
number theory Number theory | Definition, Topics, & History | Britannica 0 is not a natural number? it depends on the mathematical community, if they design the norm(standard) that 0 i a natural number, then it is. but in a default condition, we normally set 0 is not natural number. b|a Vertical bar sign in Discrete mathematics, means b divides a, b is a factor of a. division theory ? Euclidean division - Wikipedia ...
cognitive conception completeness property 完整属性 upper and lower bound 上界、下界 least upper bound 最小上界 LUB Set theory subset relations Natural numbers, Integers, Rational numbers, Real numbers N ⊂ Z ⊂ Q ⊂ R Elements of Set Theory(SUPPLEMENT, KEITH DEVLIN: Introduction to Mathematical Thinking) empty set ∅ different from Φ(phi) the 21st letter of the Greek alphabet eg. Set A If $A(x)$ is some property, the set of all those x which satisfy $A(x)$ is denoted by {$x | A(x)$} ...
least upper bound A set A of reals can have at most one least upper bound. if a is LUB, then a≤b if b is LUB, then b≤a if a, b are both two LUBs, then b=a, such as {42} has the only element 42 as both an upper bound and a lower bound. lower bound if a set A of reals has a lower bound, it has infinitely many lower bounds. ...