discrete_mathematics_with_applications/chapter_8/test_yourself.md

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Test Yourself

1. If R is a relation from A to B, x \in A, and y \in B, the notation x R y means that ____.

x is related to y by R

2. If R is a relation from A to B, x \in A, and y \in B, the notation x \cancel{R} y means that ____.

x is not related to y by R.

3. If R is a relation from A to B, x \in A, and y \in B, the notation (y, x) \in R^{-1} if, and only if, ____.

 (x, y) \in R 

4. A relation on a set A is a relation from ____ to ____.

A; A

5. If R is a relation on a set A, the directed graph of R has an arrow from x to y if, and only if, ____.

x is related to y by R

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Test Yourself

1. For a relation R on a set A to be reflexive means that ____.

2. For a relation R on a set A to be symmetric means that ____.

3. For a relation R on a set A to be transitive means that ____.

4. To show that a relation R on an infinite set A is reflexive, you suppose that ____ and you show that ____.

5. To show that a relation R on an infinite set A is symmetric, you suppose that ____ and you show that ____.

6. To show that a relation R on an infinite set A is transitive, you suppose that ____ and you show that ____.

7. To show that a relation R on a set A is not reflexive, you ____.

8. To show that a relation R on a set A is not symmetric, you ____.

9. To show that a relation R on a set A is not transitive, you ____.

10. Given a relation R on a set A, the transitive closure of R is the relation R^t on A that satisfies the following three properties: ____, ____, and ____.