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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
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
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In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset P = (X,≤) is (isomorphic to) an inclusion order (just as every group is isomorphic to a permutation group – see Cayley's theorem). To see this, associate to each element x of X the set
then the transitivity of ≤ ensures that for all a and b in X, we have
There can be sets of cardinality less than such that P is isomorphic to the inclusion order on S. The size of the smallest possible S is called the 2-dimension of P.
Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice Qn, which is the collection of all 2n subsets of an n-element set, the interval-containment orders, which are precisely the orders of order dimension at most two, and the dimension-n orders, which are the containment orders on collections of n-boxes anchored at the origin. Other containment orders that are interesting in their own right include the circle orders, which arise from disks in the plane, and the angle orders.
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In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset P = (X,≤) is (isomorphic to) an inclusion order (just as every group is isomorphic to a permutation group – see Cayley's theorem). To see this, associate to each element x of X the set X ≤ ( x ) = { y ∈ X ∣ y ≤ x } ; {\displaystyle X_{\leq (x)}=\{y\in X\mid y\leq x\};} then the transitivity of ≤ ensures that for all a and b in X, we have X ≤ ( a ) ⊆ X ≤ ( b ) precisely when a ≤ b . {\displaystyle X_{\leq (a)}\subseteq X_{\leq (b)}{\text{ precisely when }}a\leq b.} There can be sets S {\displaystyle S} of cardinality less than | X | {\displaystyle |X|} such that P is isomorphic to the inclusion order on S. The size of the smallest possible S is called the 2-dimension of P. Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice Qn, which is the collection of all 2n subsets of an n-element set, the interval-containment orders, which are precisely the orders of order dimension at most two, and the dimension-n orders, which are the containment orders on collections of n-boxes anchored at the origin. Other containment orders that are interesting in their own right include the circle orders, which arise from disks in the plane, and the angle orders.
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words, it is a collection of subsets of X {\displaystyle X} that satisfies the definition of a filter on X {\displaystyle X} and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X {\displaystyle X} that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X {\displaystyle X} can also be characterized as a filter on X {\displaystyle X} with the property that for every subset A {\displaystyle A} of X {\displaystyle X} either A {\displaystyle A} or its complement X ∖ A {\displaystyle X\setminus A} belongs to the ultrafilter. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set P ( X ) {\displaystyle {\mathcal {P}}(X)} and the partial order is subset inclusion ⊆ . {\displaystyle \,\subseteq .} This article deals specifically with ultrafilters on a set and does not cover the more general notion. There are two types of ultrafilter on a set. A principal ultrafilter on X {\displaystyle X} is the collection of all subsets of X {\displaystyle X} that contain a fixed element x ∈ X {\displaystyle x\in X} . The ultrafilters that are not principal are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exist models of ZF where every ultrafilter on a set is principal. Ultrafilters have many applications in set theory, model theory, and topology. Usually, only free ultrafilters lead to non-trivial constructions. For example, an ultraproduct modulo a principal ultrafilter is always isomorphic to one of the factors, while an ultraproduct modulo a free ultrafilter usually has a more complex structure.
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove the statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument:Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} suppose that a is a particular but arbitrarily chosen element of A show that a is an element of B. The validity of this technique can be seen as a consequence of universal generalization: the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c. Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above.
In Boolean algebra, the inclusion relation a ≤ b {\displaystyle a\leq b} is defined as a b ′ = 0 {\displaystyle ab'=0} and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order. The inclusion relation a < b {\displaystyle a
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