[Adapted from Rosen 6/e]
State whether each of the following relationships between
sets is true or false. Justify your answer briefly.
(Here, ∅ is the empty set.)
∅ ∈ ∅
{∅} ∈ {∅}
∅ ∈ {∅}
{∅} ⊆ {∅}
∅ ⊆ ∅
[Adapted from Epp 3/e]
Let R be the set of all real numbers and let
A indicate
the complement of the set A. We define the sets A,
B and C as follows:
A = { x ∈ R | -3 ≤ x ≤ 0 }
B = { x ∈ R | -1 < x < 2 }
C = { x ∈ R | 6 < x ≤ 8 }
Describe the following sets:
A ∪ B
A ∩ C
A
∪
B
A ∩
B
A ∪ B
For each of the following functions, state whether the function
is one-to-one, whether the function is onto and whether the function
is a bijection. Pay close attention to the domain and codomain
of each function. Briefly justify your answer.
f : N → N,
f (n) = n^{2} + 5.
f : Z → Z,
f (n) = n^{2} + 5.
f : N → N,
f (n) = 2 n + 7.
f : R → R,
f (x) = 2 x + 7.
f : R → R,
f (x) = 2 x^{3} + 1.
Note: N, Z and R denote the set of natural numbers,
integers and real numbers respectively.