2.9 Partial Orders 2.9 Partial Orders By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. 2.9 Partial Orders By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. So a partial order ≤ on a set X is a binary relation on X such that: 2.9 Partial Orders By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. So a partial order . on a set X is a binary relation on X such that: • a . a for all a 2 X (reflexive) 2.9 Partial Orders By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. So a partial order . on a set X is a binary relation on X such that: • a . a for all a 2 X (reflexive) • for all a; b 2 X, if a . b and b . a then a = b (antisymmetric) 2.9 Partial Orders By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. So a partial order . on a set X is a binary relation on X such that: • a . a for all a 2 X (reflexive) • for all a; b 2 X, if a . b and b . a then a = b (antisymmetric) • for all a; b; c 2 X, if a . b and b . c then a . c (transitive). Examples. Examples. • Any set of real numbers X equipped with the usual ≤. Examples. • Any set of real numbers X equipped with the usual ≤. In fact this order is total meaning that for all x; y 2 X, either x ≤ y or y ≤ x (or possibly both!). Examples. • Any set of real numbers X equipped with the usual ≤. In fact this order is total meaning that for all x; y 2 X, either x ≤ y or y ≤ x (or possibly both!). • Let X = 2Y where Y is a set. This means X is all subsets of Y . Examples. • Any set of real numbers X equipped with the usual ≤. In fact this order is total meaning that for all x; y 2 X, either x ≤ y or y ≤ x (or possibly both!). • Let X = 2Y where Y is a set. This means X is all subsets of Y . Then the inclusion relation ⊆ given by S ⊆ T if S is a subset of T is a partial order on X. Examples. • Any set of real numbers X equipped with the usual ≤. In fact this order is total meaning that for all x; y 2 X, either x ≤ y or y ≤ x (or possibly both!). • Let X = 2Y where Y is a set. This means X is all subsets of Y . Then the inclusion relation ⊆ given by S ⊆ T if S is a subset of T is a partial order on X. This partial order is not total: e.g. f1; 2g and f2; 3g are subsets of Y = f1; 2; 3g but neither contains the other. • The relation of divisibility on the natural numbers N is a partial order: • The relation of divisibility on the natural numbers N is a partial order: mjn if m is a factor of n, i.e. if n = km for some k 2 N. • The relation of divisibility on the natural numbers N is a partial order: mjn if m is a factor of n, i.e. if n = km for some k 2 N. Again this partial order is not total • The relation of divisibility on the natural numbers N is a partial order: mjn if m is a factor of n, i.e. if n = km for some k 2 N. Again this partial order is not total (2 does not divide 3 and 3 does not divide 2). A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). But digraphs of partial orders are usually drawn in a different way: A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram. A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram. For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram. For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, and then draw b higher than a, and omit the arrow. A set X together with a partial order . on it is called a partially or- dered set (X; .), or a poset. If X is finite, there is a digraph associated with the poset (X; .). But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram. For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, and then draw b higher than a, and omit the arrow. The Hasse diagram contains complete information about .. For example, let X = f1; 2; 3; 4g with the partial order the usual ordering of real numbers: For example, let X = f1; 2; 3; 4g with the partial order the usual ordering of real numbers: 1 ≤ 2 ≤ 3 ≤ 4: For example, let X = f1; 2; 3; 4g with the partial order the usual ordering of real numbers: 1 ≤ 2 ≤ 3 ≤ 4: The digraph would look like: For example, let X = f1; 2; 3; 4g with the partial order the usual ordering of real numbers: 1 ≤ 2 ≤ 3 ≤ 4: The digraph would look like: 1 • • 2 4 • • 3 but the Hasse diagram is as follows: but the Hasse diagram is as follows: • 4 • 3 • 2 • 1 Another example: X = f1; 2; 3; 4; 5; 6g, ordered by divisibility: Another example: X = f1; 2; 3; 4; 5; 6g, ordered by divisibility: 1jn for all n 2 X; 2j2; 4; 6; 3j3; 6; 4j4; 5j5; 6j6: Another example: X = f1; 2; 3; 4; 5; 6g, ordered by divisibility: 1jn for all n 2 X; 2j2; 4; 6; 3j3; 6; 4j4; 5j5; 6j6: The Hasse diagram is as follows: Another example: X = f1; 2; 3; 4; 5; 6g, ordered by divisibility: 1jn for all n 2 X; 2j2; 4; 6; 3j3; 6; 4j4; 5j5; 6j6: The Hasse diagram is as follows: 4 6 • • 5 • 2 • • 3 • 1 A final example: A final example: X = 2f1;2g = f;; f1g; f2g; f1; 2gg; A final example: X = 2f1;2g = f;; f1g; f2g; f1; 2gg; ordered by inclusion. A final example: X = 2f1;2g = f;; f1g; f2g; f1; 2gg; ordered by inclusion. Hasse diagram: A final example: X = 2f1;2g = f;; f1g; f2g; f1; 2gg; ordered by inclusion. Hasse diagram: f1; 2g f1g f2g f0g Equivalence relations and partial orders are examples of preorders: Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. So R∗ for any binary relation R on X is a preorder of X. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. So R∗ for any binary relation R on X is a preorder of X. Preorders can be thought of as “partial orders on the equivalence classes of a set”. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. So R∗ for any binary relation R on X is a preorder of X. Preorders can be thought of as “partial orders on the equivalence classes of a set”. An example is R defined on all people by: Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. So R∗ for any binary relation R on X is a preorder of X. Preorders can be thought of as “partial orders on the equivalence classes of a set”. An example is R defined on all people by: xRy if and only if x is at least as tall as y. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. So R∗ for any binary relation R on X is a preorder of X.