Civil Rights
Movements, leaders, victories and the continuing fight for equality.
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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Birkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff:
It may also refer to theorems named for his son, Garrett Birkhoff:
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Birkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff: Birkhoff's theorem (relativity) Birkhoff's theorem (electromagnetism) Birkhoff's ergodic theorem It may also refer to theorems named for his son, Garrett Birkhoff: Birkhoff–von Neumann theorem for doubly stochastic matrices Birkhoff's HSP theorem, concerning the closure operations of homomorphism, subalgebra and product Birkhoff's representation theorem for distributive lattices Birkhoff's theorem (equational logic), stating that syntactic and semantic consequence coincide
In logic, the symbol ⊨, ⊧ or ⊨ {\displaystyle \models } is called the double turnstile. It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than". It is closely related to the turnstile symbol ⊢ {\displaystyle \vdash } , which has a single bar across the middle, and which denotes syntactic consequence (in contrast to semantic).
In logic, Birkhoff's theorem in equational logic states that an equality t = u is a semantic consequence of a set of equalities E, if and only if t = u can be proven from the set of equalities. It is named after Garrett Birkhoff.
In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, also called the duality principle. It is the most widely known example of duality in logic. The duality consists in these metalogical theorems: In classical propositional logic, the connectives for conjunction and disjunction can be defined in terms of each other, and consequently, only one of them needs to be taken as primitive. If φ D {\displaystyle \varphi ^{D}} is used as notation to designate the result of replacing every instance of conjunction with disjunction, and every instance of disjunction with conjunction (e.g. p ∧ q {\displaystyle p\land q} with q ∨ p {\displaystyle q\lor p} , or vice-versa), in a given formula φ {\displaystyle \varphi } , and if φ ¯ {\displaystyle {\overline {\varphi }}} is used as notation for replacing every sentence-letter in φ {\displaystyle \varphi } with its negation (e.g., p {\displaystyle p} with ¬ p {\displaystyle \neg p} ), and if the symbol ⊨ {\displaystyle \models } is used for semantic consequence and ⟚ for semantical equivalence between logical formulas, then it is demonstrable that φ D {\displaystyle \varphi ^{D}} ⟚ ¬ φ ¯ {\displaystyle \neg {\overline {\varphi }}} , and also that φ ⊨ ψ {\displaystyle \varphi \models \psi } if, and only if, ψ D ⊨ φ D {\displaystyle \psi ^{D}\models \varphi ^{D}} , and furthermore that if φ {\displaystyle \varphi } ⟚ ψ {\displaystyle \psi } then φ D {\displaystyle \varphi ^{D}} ⟚ ψ D {\displaystyle \psi ^{D}} . (In this context, φ ¯ D {\displaystyle {\overline {\varphi }}^{D}} is called the dual of a formula φ {\displaystyle \varphi } .)
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →The Greenwood District.