Lecture 5 - Relations - Relations(Computer Science Notes...

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Relations (Computer Science Notes) Relations A relation R on a set A is a subset of A × A R is a set of ordered pairs of elements from A. If R contains the pair (x, y), we say that x is related to y, or xRy in shorthand. We’ll write x/Ry to mean that x is not related to y For example, suppose we let A = {2, 3, 4, 5, 6, 7, 8}. We can define a relation W on A by xWy if and only if x ≤ y ≤ x + 2. Then W contains pairs like (3, 4) and (4, 6), but not the pairs (6, 4) and (3, 6). Under this relation, each element of A is related to itself. So W also contains pairs like (5, 5) We can define another relation S on A by saying that xSy is in S if x ≡ y (mod 3) X Y Properties of Relations: Reflexive Reflexive: every element is related to itself, elements related to themselves have self- loops Irreflexive: no element is related to itself Neither reflexive nor irreflexive: some elements are related to themselves but some aren’t The familiar relations ≤ and = on the real numbers are reflexive, but < is irreflexive R is reflexive if xRx for all x A R is irreflexive if x/Rx for all x A not reflexive: there is an x A, x/Ry
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