Math 121: Algebra and Trigonometry
Problem Session I: Thursday January 26
22/23
These exercises are designed to get you to focus on what you are reading, so you
understand the definitions properly. Pay attention to the material; definitions
are very important.1
1. Which of the following sets are empty? Justify your answers.
(a) {x ∈ R : x2 = 9 and 3x = 12}
(d) {z ∈ Z : 1 < z < 2}
(b) {x ∈ R : x 6= x}
(e) {n ∈ N : n2 = 5}
(c) {y ∈ R : 1 < y < 3}
(f) {n ∈ R : n2 = 5}
2. Perform the indicated operations on the following intervals and write down the resulting
sets.
(a) [1, 2] ∪ [2, 3]
(e) [1, 4] \ (2, 3)
(b) [1, 2] ∩ [2, 3]
(f) [1, 4] \ [2, 3]
(c) (1, 2) ∩ (2, 3)
(g) [1, 2] ∪ [3, 4]
(d) [1, 3] \ [2, 3]
(h) [0, 1]c
3. Determine whether each of these statements is true or false.
(a) x ∈ {x}
(e) 0 ∈ ∅
(b) {x} ∈ {x}
(f) ∅ ∈ {0}
(c) {x} ∈ {{x}}
(g) 10 6∈ (−∞, π 2 )
(d) ∅ ∈ {x}
(h) π ∈ (2, ∞)
4. For each of the following sets, determine whether
• 2
• {2}
is an element of that set.
1
(a) {{2}, {2, {2}}}
(c) {{{2}}}
(b) {2, {2}}
(d) {{2}, {{2}}}
Most of these exercises are courtesy of Dr. B.V. Normenyo
1
5. List the members of the following sets
(a) {y ∈ R|x2 = 1} Pay attention!
(d) {w ∈ Q|w is a factor of 12}
(b) {x ∈ Z+ |x < 12}
(e) {2 + (−1)n |n ∈ N}
(c) {a ∈ Z|a2 = 2}
(f) {q 2 − q|q = 0, 1, 2, 3, 4}
6. Simplify the following expressions:
−1 4x−2 (yz)−1
3x
(b)
(a)
4y −1
23 x4 y
(c)
7. Simplify the following radical expressions:
s
√
√
2
(c)
−
18
+
2
8
3xy
(a) 3
√
√
4
81x4 y 2
(d) 4 32x + 2x5
√
√
(b) 2 12 − 3 27
8. Rationalize the following denominators:
√
2
(a) √
7+2
√
2− 5
√
(b)
2+3 5
2
√
3−1
(c) √
2 3+3
√
√
x+h− x
(d) √
√
x+h+ x
5x−2
6y −2
−3