CSCI 301 L18 Notes

Lecture 18 - Notes

Relations

• Know the definition of a relation on a set.
• Be able to apply the definitions of the following properties of relations:
  • Reflexive
  • Symmetric
  • Transitive

Announcements

• Midterm exam details - in case you missed it, see the announcements from the L17 Notes

• Current plan: continuing Wednesday release/due date for homeworks all the way through the end of the quarter. This means there is an assignment due on Wednesday of the last week of classes.

Relations

Example:

Let \(A = \{1, 2, 3\}\). The relation \(R\) meaning “greater than equal to” is a subset of \(A \times A\): \[ R = \{(1, 1), (2, 1), (2, 2), (3, 3), (3, 2), (3, 1)\} \subseteq A \times A \] We can also draw a diagram representing this relation:

where each element of \(A\) is a dot, and each ordered pair in the relation is drawn as an arrow from its first element to its second element.

Do Exercises Part A

Properties of Relations

Consider \(R\) from above. Is it:

• Reflexive? Yes! \(a \ge a\) for all \(a \in \{1, 2, 3\}\).
• Symmetric? No! \(1 \ge 3\) but \(3 \not\ge 1\).
• Transitive? Yes! This is trickier to check:
  • \(3 \ge 2\) and \(2 \ge 1\), but \(3 \ge 1\)
  • \(3 \ge 3\) and \(3 \ge 2\), but \(3 \ge 2\)
  • …and so on

Do Exercises Part B