A relation R on a set A is symmetric iff aRb implies that bRa, for every a,b ε A. Think $\le$. How can a relation be symmetric and anti-symmetric? 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}. (2,1) is not in B, so B is not symmetric. Antisymmetric relation is a concept of set theory that builds upon both symmetric and asymmetric relation in discrete math. Whether the wave function is symmetric or antisymmetric under such operations gives you insight into whether two particles can occupy the same quantum state. 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. the truth holds vacuously. How can a relation be symmetric an anti symmetric? A relation can be both symmetric and antisymmetric. 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. 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? Note: Asymmetric is the opposite of symmetric but not equal to antisymmetric. Symmetric or antisymmetric are special cases, most relations are neither (although a lot of useful/interesting relations are one or the other). 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. 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 means that the only way for both $aRb$ and $bRa$ to hold is if $a = b$. Or we can say, the relation R on a set A is asymmetric if and only if, (x,y)∈R (y,x)∉R. ? 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. Since (1,2) is in B, then for it to be symmetric we also need element (2,1). Yes. A relation can be neither symmetric nor antisymmetric. – 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 – … Question: How Can A Matrix Representation Of A Relation Be Used To Tell If The Relation Is: Reflexive, Irreflexive, Symmetric, Antisymmetric, Transitive? It can be reflexive, but it can't be symmetric for two distinct elements. Apply it to Example 7.2.2 to see how it works. It is an interesting exercise to prove the test for transitivity. 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. This question hasn't been answered yet Ask an expert.