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Movements, leaders, victories and the continuing fight for equality.
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THE BHP WIRE —
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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.
Innovation, patents, science, technology and world-changing contributions.
Pioneers, champions, Negro Leagues, records, activism and excellence.
Meet the people whose lives, choices and achievements shaped the journey.
Black towns, communities, institutions and places where history happened.
Moments that changed communities, movements, institutions and the nation.
Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
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
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given a function f : A → B {\displaystyle f:A\to B} , the image of an element a ∈ A {\displaystyle a\in A} is the element f ( a ) ∈ B {\displaystyle f(a)\in B} in the codomain. The pre-image of an element b ∈ B {\displaystyle b\in B} is any element a ∈ A {\displaystyle a\in A} in the domain such that f ( a ) = b {\displaystyle f(a)=b} . Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if and only if it is invertible; that is, a function f : X → Y {\displaystyle f:X\to Y} is bijective if and only if there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f, such that each of the two ways for composing the two functions produces an identity function: g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for each x {\displaystyle x} in X {\displaystyle X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for each y {\displaystyle y} in Y . {\displaystyle Y.} For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function. A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective. The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...), up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group. Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutations, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function f : X → Y {\displaystyle f\colon X\to Y} : The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection. Notationally: ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⟹ x = x ′ , {\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',} or, equivalently (using logical transposition), ∀ x , x ′ ∈ X , x ≠ x ′ ⟹ f ( x ) ≠ f ( x ′ ) . {\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').} The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain; that is, if the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y , ∃ x ∈ X , y = f ( x ) . {\displaystyle \forall y\in Y,\exists x\in X,y=f(x).} The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain; that is, if the function is both injective and surjective. A bijective function is also called a bijection. That is, combining the definitions of injective and surjective, ∀ y ∈ Y , ∃ ! x ∈ X , y = f ( x ) , {\displaystyle \forall y\in Y,\exists !x\in X,y=f(x),} where ∃ ! x {\displaystyle \exists !x} means "there exists exactly one x". In any case (for any function), the following holds: ∀ x ∈ X , ∃ ! y ∈ Y , y = f ( x ) . {\displaystyle \forall x\in X,\exists !y\in Y,y=f(x).} An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.
In mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set A {\displaystyle A} is generally denoted by | A | {\displaystyle \vert A\vert } , with a vertical bar on each side, though it may also be denoted by A {\displaystyle A} , card ( A ) , {\displaystyle \operatorname {card} (A),} or # A . {\displaystyle \#A.} Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. The cardinality of a finite set can be identified with a natural number, which can be found simply by counting its elements. For example, the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 4 , 5 , 6 } {\displaystyle \{4,5,6\}} both have the same cardinality 3, as evidenced by the bijection { 1 ↦ 4 , 2 ↦ 5 , 3 ↦ 6 } {\displaystyle \{1\mapsto 4,2\mapsto 5,3\mapsto 6\}} . The behavior of cardinalities of infinite sets is more complex. For example, there exists a bijection between the set of all natural numbers N {\displaystyle \mathbb {N} } and the set of all rational numbers Q {\displaystyle \mathbb {Q} } , and thus | N | = | Q | {\displaystyle \vert \mathbb {N} \vert =\vert \mathbb {Q} \vert } even though N {\displaystyle \mathbb {N} } is a proper subset of Q {\displaystyle \mathbb {Q} } —something that cannot happen with proper subsets of finite sets. However, a fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers R {\displaystyle \mathbb {R} } is greater than the cardinality of N {\displaystyle \mathbb {N} } . The cardinality of N {\displaystyle \mathbb {N} } is usually denoted by ℵ 0 {\displaystyle \aleph _{0}} (aleph-null), since it is the smallest aleph number. The properties of other aleph numbers and of infinite cardinal numbers in general depend on statements independent of Zermelo–Fraenkel set theory, such as the axiom of choice and the continuum hypothesis. For example, all infinite cardinal numbers are aleph numbers if and only if the axiom of choice is true. Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 and 7; however, the integers 2 and 3 are not because each can only be divided by one and itself. The first 25 composite numbers (all up to 36) are: Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 23 × 32 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, which do not necessarily reveal the factorization of a composite input. Grimm's conjecture states that, for every set of consecutive composite numbers, there is an equally sized set of prime numbers, and a bijection mapping each composite in the former set to a prime in the latter set that it is divisible by.
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 →Madam C.J. Walker