Think [math]\le[/math]. Yes. Whether the wave function is symmetric or antisymmetric under such operations gives you insight into whether two particles can occupy the same quantum state. A relation R on a set A is antisymmetric iff aRb and bRa imply that a = b. Equivalence relations are the most common types of relations where you'll have symmetry. – antisymmetric states 㱺 fermions half-integer spin • Pauli from properties of electrons in atoms – symmetric states 㱺 bosons integer spin • Considerations related to electromagnetic radiation (photons) • Can also consider quantization of “field” equations – … ? Symmetry Properties of Relations: A relation {eq}\sim {/eq} on the set {eq}A {/eq} is a subset of the Cartesian product {eq}A \times A {/eq}. Given that P ij 2 = 1, note that if a wave function is an eigenfunction of P ij , then the possible eigenvalues are 1 and –1. It can be reflexive, but it can't be symmetric for two distinct elements. Also, i'm curious to know since relations can both be neither symmetric and anti-symmetric, would R = {(1,2),(2,1),(2,3)} be an example of such a relation? Symmetric or antisymmetric are special cases, most relations are neither (although a lot of useful/interesting relations are one or the other). How can a relation be symmetric and anti-symmetric? This question hasn't been answered yet Ask an expert. At its simplest level (a way to get your feet wet), you can think of an antisymmetric relation of a set as one with no ordered pair and its reverse in the relation. For example: If R is a relation on set A = {12,6} then {12,6}∈R implies 12>6, but {6,12}∉R, since 6 is not greater than 12. A relation can be neither symmetric nor antisymmetric. A relation can be both symmetric and antisymmetric. Antisymmetric relation is a concept of set theory that builds upon both symmetric and asymmetric relation in discrete math. A relation R on a set A is symmetric iff aRb implies that bRa, for every a,b ε A. Or we can say, the relation R on a set A is asymmetric if and only if, (x,y)∈R (y,x)∉R. Apply it to Example 7.2.2 to see how it works. (2,1) is not in B, so B is not symmetric. How can a relation be symmetric an anti symmetric? Question: How Can A Matrix Representation Of A Relation Be Used To Tell If The Relation Is: Reflexive, Irreflexive, Symmetric, Antisymmetric, Transitive? Antisymmetric means that the only way for both [math]aRb[/math] and [math]bRa[/math] to hold is if [math]a = b[/math]. Thus, it will be never the case that the other pair you're looking for is in $\sim$, and the relation will be antisymmetric because it can't not be antisymmetric, i.e. If a relation \(R\) on \(A\) is both symmetric and antisymmetric, its off-diagonal entries are all zeros, so it is a subset of the identity relation. Note: Asymmetric is the opposite of symmetric but not equal to antisymmetric. Since (1,2) is in B, then for it to be symmetric we also need element (2,1). It is an interesting exercise to prove the test for transitivity. the truth holds vacuously.

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