In logic and mathematics, [[anti-symmetry]] refers to a particular property that a relation or operation may possess. In the context of relations, a binary relation R on a set A is said to be anti-symmetric if for every pair of distinct elements x and y in A, if (x, y) belongs to R and (y, x) also belongs to R, then x and y must be the same element. In other words, if there is a directed edge from x to y and another directed edge from y to x in the relation R, then x must be equal to y. For example, consider the relation "is less than or equal to" (≤) on the set of integers. This relation is anti-symmetric because if x ≤ y and y ≤ x, then it implies that x = y. However, the relation "is less than" (<) on the same set is not anti-symmetric because there exist pairs of distinct integers where both x < y and y < x are true. In mathematics, anti-symmetry can also apply to certain operations or functions. For instance, an operation such as subtraction (-) is anti-symmetric because for any two distinct numbers a and b, if a - b = 0 and b - a = 0, then it follows that a = b. Anti-symmetry plays an important role in various mathematical proofs and reasoning. It helps establish uniqueness properties within relations or operations by ensuring that no distinct elements have both directions of connection simultaneously. # References ```dataview Table title as Title, authors as Authors where contains(subject, "anti-symmetric") ```