Stonybrook University CSE Relations Strings Worksheet – Description
Let A be the set of all strings of length 6 consisting of x’s and y’s. Then A is denoted Σ6 where Σ = {x, y}. Define a binary relation R from A to A as follows:
For all strings s and t in A, s R t ⇔ the first four characters of s equal the first four characters of t.
a) Is xxyxyx R xxxyxy ?
b) Is yxyyyx R yxyyxy?
c) Is xyxxxx R yxxxxx?
Let A = { 2, 3, 4 } and B = { 6, 8, 10 } and define a binary relation R from A to B as follows:
for all (x, y) ∈ A X B, (x, y) ∈ R ⇔ x | y.
a) Is 4 R 6?
b) Is 4 R 8?
c) Is (3, 8) ∈ R? Is (2, 10) ∈ R?
d) Write R as a set of ordered pairs.
Prove that for all integers m and n, m – n is even if and only if both m and n are even or both m and n are odd.
Declare a binary relation S from R to R as:
for all (x, y) ∈ R X R, x S y ⇔ x ≥ y.
Draw the graph of S in the Cartesian plane.
Define a binary relation T from R to R as follows:
for all (x, y) ∈ R X R, x T y ⇔ y = x2.
Draw the graph of T in the Cartesian plane.
For the following relations, 1) draw the directed graph, 2) determine whether it is reflexive, symmetric, and transitive. Give a counterexample in each case in which the relation does not satisfy one of these properties.
a) R1 = {(0, 0), (0, 1), (0, 3), (1, 1), (1, 0), (2, 3), (3, 3)}
b) R2 = {(2, 3), (3, 2)}
c) R3 = {(0, 1), (0, 2)}
D is the binary relation defined on R as follows::
for all x, y ∈ R, x D y ⇔ xy ≥ 0.
a) Draw a Cartesian Graph of the relation.
b) Is it Reflexive?
c) Is it Symmetric?
d) Is it Transitive?
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