What Is A Relation? Example. is a relation from A to B.
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1 3.3 Relations
2 What Is A Relation? Let A and B be nonempty sets. A relation R from A to B is a subset of the Cartesian product A B. If R A B and if (a, b) R, we say that a is related to b by R and we write a R b.
3 What Is A Relation? Let A and B be nonempty sets. A relation R from A to B is a subset of the Cartesian product A B. If R A B and if (a, b) R, we say that a is related to b by R and we write a R b. Example Let A = {1, 2, 3} and let B = {r, s}. Then R = {(1, r), (2, s), (3, r)} is a relation from A to B.
4 Examples Example Let A and B be sets of real numbers. We define the relation R equals from A to B as a R b a = b
5 Examples Example Let A and B be sets of real numbers. We define the relation R equals from A to B as a R b a = b Example Let A = {1, 2, 3, 4}. Define the relation R (less than) as follows: a R b a < b Then R = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
6 Examples Example Let A be the set of positive integers. Define the following relation on A a R b a b Then 4 R 12 but 5 R 7.
7 Examples Example Let A be the set of positive integers. Define the following relation on A a R b a b Then 4 R 12 but 5 R 7. Example Let A be the set of all people in the world. We define the following relation on A: a R b if and only if there is a sequence a 0, a 1,... a n of people such that a = a 0, b = a n and a i 1 knows a i for i = 1, 2,... n.
8 Symmetry Anyone remember what the symmetric property does for us?
9 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R.
10 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R. A relation R is said to be asymmetric if for (x, y) R then (y, x) R.
11 Symmetry Anyone remember what the symmetric property does for us? A relation R is said to be symmetric if for (x, y) R then (y, x) R. A relation R is said to be asymmetric if for (x, y) R then (y, x) R. From our prior examples, can you think of any that were symmetric? Asymmetric?
12 Transitivity Does anyone remember what the transitive property does for us?
13 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c.
14 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c. In terms of the Cartesian products, we have that the relation R is transitive if for (x, y) R and (y, z) R that (x, z) R.
15 Transitivity Does anyone remember what the transitive property does for us? We say that a relation R is transitive if a R b and b R c implies a R c. In terms of the Cartesian products, we have that the relation R is transitive if for (x, y) R and (y, z) R that (x, z) R. Any of our examples transitive?
16 The Reflexive Property And how about the reflexive property - what does this property do for us?
17 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain.
18 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain. A relation R that is asymmetric, transitive and reflexive is called a partial order.
19 The Reflexive Property And how about the reflexive property - what does this property do for us? A relation R is said to be reflexive if (x, x) R for all x in the domain. A relation R that is asymmetric, transitive and reflexive is called a partial order. A relation R that is symmetric, transitive and reflexive is called an equivalence relation.
20 Inverses The inverse of a relation R, denoted R 1 is the set {(y, x) (x, y) R}
21 Inverses The inverse of a relation R, denoted R 1 is the set {(y, x) (x, y) R} Example For the relation we defined on the set A = {1, 2, 3, 4} where a R b a < b we have a R 1 b a b
22 Visual Representations - Digraphs A digraph, or directed graph, is a graph, or set of nodes connected by edges, where the edges have a direction associated with them.
23 Visual Representations - Digraphs A digraph, or directed graph, is a graph, or set of nodes connected by edges, where the edges have a direction associated with them. We can use digraphs to give visual representations of relations.
24 Visual Representations - Digraphs Example Give the relation represented by the digraph
25 Visual Representations - Digraphs Example Give the relation represented by the digraph R = {(1, 1), (1, 2), (1, 3), (2, 3)}
26 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}
27 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 2), (2, 3), (2, 4), (3, 3), (4, 2)}
28 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}
29 Visual Representations - Digraphs Example Give a visual representation for the relation R on A = {1, 2, 3, 4} where R = {(1, 1), (1, 2), (2, 2), (2, 3), (2, 4), (3, 4), (4, 1)}
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