Answer to Question #303285 in Discrete Mathematics for Justin

Question

Consider a relation R on a set A = { 2, 4, 7 }.

Given the relation R = { (2, 2), (2, 4), (2, 7), (4, 7}. Find:

1. Complement of a Relation

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2. Inverse of a Relation

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3. Composite Product R o S and S o R ; S = { (1, 2), (2, 4), (2, 7) }

Answer 1

Given $R = \{(2, 2), (2, 4), (2, 7), (4, 7)\}$ is a relation on the set $A = \{2,4,7\}$.

A relation on a set A is a subset of the cartesian product $A \times A$.

Here, $A \times A = \{(2,2),(2,4), (2,7), (4,2), (4,4), (4,7), (7,2), (7,4),(7,7)\}$

1. Complement of R = The set of all elements in $A \times A$ but not in $R$ $= \{(4,2), (4,4), (7,2), (7,4), (7,7)\}$

2. Inverse of R = $R^{-1} = \{(b,a)\mid (a,b)\in R\}=\{(2,2), (4,2),(7,2), (7,4)\}$

3. Given, $S = \{ (1, 2), (2, 4), (2, 7) \}$

$R \circ S = \{(1,2), (1,4), (1,7), (2,7)\}\\ S \circ R = \{(2,4), (2,7)\}$