Homework # 15
Due
Name ___________________
|
ℕ = set of all nonnegative integers (note that 0 ∈ ℕ) ℤ= set of all integers ℚ= set of all rational numbers ℝ= set of all real numbers ℂ= set of all complex numbers |
1. S = {a,b,c} What is the Cartesian product of the set S with itself? (i.e. what is S X S?)
2. For each of the following binary relations R on ℕ decide which of the given ordered pairs belong to R.
a. x R y ↔x = y + 1: (2,2), (2,3), (3,3), (3,2)
b. x R y ↔x divides y; (2,4), (2,5), (2,6)
c. x R y ↔x is odd; (2,3), (3,4), (4,5), (5,6)
3. Let S = { 1,2,3}
a. If a relation R on S is reflexive, what ordered pairs must belong to R?
b. If a relation R on S is symmetric
and if (a,b) ∈R,
then what other ordered pair must
belong
to R?
c. If a relation R
on S is
antisymmetric and if (a,b) and (b,a) belong to R,
what must be
true?
4. Let S = { 0,1,2,4,6} Test the
following binary relations on S for reflexivity, symmetry, antisymmetry and
transitivity. Tell why the relation R
is not reflexive,
symmetric, antisymmetric, or transitive if it isn't.
a. R = { (0,0), (1,1), (2,2),
(4,4), (6,6), (0,1), (1,2), (2,4), (4,6) }
b. R = { (0,1), (1,0), (2,4),
(4,2), (4,6), (6,4) }
5. Give an example of
a binary relation R on set S = {1,2,3} that is
neither reflexive nor irreflexive.
6. Give an example of a binary relation R on set S = {1,2,3} that is neither symmetric nor asymmetric.