Assignment 01 Relations and Their Properties

Assignment 01 NAME:_______________________________________________
Relations and Their Properties 
Directions: Show ALL work for credit. There are 5 questions. Write on your own paper.
Each part is worth 3 points, unless stated otherwise. 40 points total. You may type or
neatly write your solutions. Make sure you write your name on all papers that you use.
Scan this page at the front of your work, and compile as ONE .pdf file. Check that all work
was saved and scanned legibly.
Save your file as: A01xyLASTNAME.pdf. (where “xy” is your first and middle initial)
Once completed, attach your file under “Assignment 01” on Canvas. Thank you!
1) For the relation 𝑅𝑅 = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} on the set
𝐴𝐴 = {1, 2, 3, 4} , explain/show whether or not the relation is the following:
(For any credit, be sure to give a reason why for each). (2 points each)
(a) reflexive,
(b) symmetric,
(c) antisymmetric,
(d) transitive.
2) Let the sets be relations on the real numbers: 𝑅𝑅1 = {(𝑎𝑎, 𝑏𝑏) ∈ ℝ2| 𝑎𝑎 ≥ 𝑏𝑏}, the
“greater than or equal to” relation and let 𝑅𝑅2 = {(𝑎𝑎, 𝑏𝑏) ∈ ℝ2| 𝑎𝑎 ≠ 𝑏𝑏}, the “unequal
to” relation.
Find:
(a) 𝑅𝑅1 ∩ 𝑅𝑅2 (write out the relation in the set notation, as 𝑅𝑅1 and 𝑅𝑅2 were written)
(b) 𝑅𝑅1 − 𝑅𝑅2 (write out the relation in the set notation, as 𝑅𝑅1 and 𝑅𝑅2 were written)
(c) 𝑅𝑅1⨁𝑅𝑅2 (write out the relation in the set notation, as 𝑅𝑅1 and 𝑅𝑅2 were written)
3)(a) How many binary relations are there on the set {𝑎𝑎, 𝑏𝑏, 𝑐𝑐}? (2 points)
(b) If 𝑅𝑅 = {(1, 1), (1, 2), (2, 4), (3, 1), (3, 0)} , 𝑆𝑆 = {(1, 2), (2, 0), (3, 1), (0, 0), (4, 3)}
find 𝑆𝑆 ∘ 𝑅𝑅, with elements listed as above.
4) 𝑅𝑅 is the relation represented by the matrix 𝑴𝑴𝑅𝑅 = �
1 0 0
1 1 1
0 1 0
�, find the matrix for:
(a) 𝑅𝑅−1
(b) �𝑅𝑅���
(c) 𝑅𝑅 ∘ 𝑅𝑅 (i.e. 𝑅𝑅2)
5) (a) The relation R is on {1, 2, 3}. Represent the relation (4 points)
𝑅𝑅 = {(1, 1), (2, 1), (2, 2), (2, 3), (3, 2)} with a matrix.
(b) By looking at the matrix, is the relation R reflexive? Why or why not? (2 points)
(c) Draw the directed graph that represents the relation R. (3 points)