Reading Quiz Section 7.1
1. A relation R ⊆ A × B is
(a) a nonempty subset of A × B
(b) a proper subset of A × B
(c) a function from A to B
(d) a subset of A × B
2. If A ⊆ R, then the graph of a symmetric relation R ⊆ A × A has what kind of symmetry?
(a) reflection symmetry across the x-axis
(b) reflection symmetry across the y-axis
(c) reflection symmetry across the line y = x
(d) symmetry across the origin
3. True or False: if R is symmetric, then it must contain an even number of elements.
Practice Problems Section 7.1
1. Given constants a, b, c, let L
a,b,c
=
(x, y) : ax + by = c
⊆ R
2
.
(a) Describe L
a,b,c
geometrically.
(b) Let A = R
2
and B =
L
a,b,c
: a, b, c ∈ R
. Define R ⊆ A × B by
(x, y) R L
a,b,c
⇐⇒ ax + by = c
Determine whether each of the following is true or false.
i. ( 1, 0) R L
1,1,1
ii. ( 3, −2) R L
1,1,1
iii. If (x, y) R L
a,b,c
and (x, y) R L
d,e, f
for some (x, y) then L
a,b,c
= L
d,e, f
iv. Suppose (x, y) R L
a,b,c
. Then there exists d, e, f ∈ R such that
(x, y) R L
d,e, f
and L
a,b,c
∩ L
d,e, f
= ∅
2. Let X be a set. Let R ⊆ P(X) × P(X) be the relation A R B ⇐⇒ A ⊆ B.
(a) Show that A (R ∩ R
−1
) B implies A = B.
(b) If X = {a, b}, compute R
−1
explicitly as a set of ordered pairs.
