Symmetric difference

Summary

In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets {1,2,3} and {3,4} is {1,2,4}. The symmetric difference of the sets A and B is commonly denoted by A \ominus B or A\operatorname \vartriangle B. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Properties The symmetric difference is equivalent to the union of both relative complements, that is: :A, \vartriangle,B = \left(A \setminus B\right) \c

Official source

https://en.wikipedia.org/wiki/Symmetric_difference

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Twisting structures and strongly homotopy morphisms

Kathryn Hess Bellwald

In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad and let BP be the bar construction on the operad P. To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras. Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved. In the appendix, we study the particular case of the two-sided Koszul resolution of the associative operad.

2010

On arithmetic progressions in symmetric sets in finite field model

Jan Hazla

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set S subset of Z(q)(n) containing vertical bar S vertical bar = mu . q(n) elements must contain at least delta(q, mu) . q(n) . 2(n) arithmetic progressions x, x+d, . . . , x+(q - 1).d such that the difference d is restricted to lie in {0, 1}(n). Second, we show that for prime p a symmetric set S subset of F-p(n) with vertical bar S vertical bar = mu . p(n) elements contains at least mu(C(p)) . p(2n) arithmetic progressions of length p. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.

ELECTRONIC JOURNAL OF COMBINATORICS2020

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

2019