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Similarly, the subset order on the subsets of any given set is antisymmetric: given two sets and , if every element in also is in and every element in is also in , then and must contain all the same elements and therefore be equal: = A real-life example of a relation that is typically antisymmetric is "paid the restaurant bill of" (understood ...
Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric). The electromagnetic tensor, in electromagnetism. The Riemannian volume form on a pseudo-Riemannian manifold.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...
where the order of the tensor product matters ( if | | , then the particle 1 occupies the state n 2 while the particle 2 occupies the state n 1). This is the canonical way of constructing a basis for a tensor product space H ⊗ H {\displaystyle H\otimes H} of the combined system from the individual spaces.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
If there is already an active hint on the board, a hint will show that word’s letter order. Related: 300 Trivia Questions and Answers to Jumpstart Your Fun Game Night.
Order-dual. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse. Order-embedding. A function f between posets P and Q is an order-embedding if, for all elements x, y of P, x ≤ y (in P) is equivalent to f(x) ≤ f(y) (in Q). Order isomorphism.
The number of distinct terms () in the expansion of the determinant of a skew-symmetric matrix of order was considered already by Cayley, Sylvester, and Pfaff. Due to cancellations, this number is quite small as compared the number of terms of the determinant of a generic matrix of order n {\displaystyle n} , which is n ! {\displaystyle n!} .