Set Notation

Show your work for each problem.

1.
Which of the following is a correct way to write a set?
1. a)
2. b)
3. c)
4. d)
Answer:
is correct because sets are written with curly braces. Parentheses () are for ordered pairs, and square brackets [] are for intervals.
2.
What does mean?
1. a)5 is equal to set A
2. b)5 is not an element of set A
3. c)5 is a subset of A
4. d)5 is an element of set A
Answer: 5 is an element of set A
The symbol means 'is an element of'. So reads '5 is an element of set A' or '5 belongs to A'.
3.
How many elements are in the set ?
Answer: 5
The set contains 5 elements: a, e, i, o, u. Each vowel is counted once.
4.
Which symbol represents the empty set?
1. a)
2. b)
3. c)
4. d)
Answer:
(or ) represents the empty set with no elements. is a set containing zero, which is different!
5.
If , is an element of ? Answer yes or no.
Answer: yes
Yes, 6 is an element of A because it appears in the set . We write .
6.
Write the set in roster form.
Answer: {2, 4, 6, 8}
1. What is the first positive even number? 2
2. List all positive even numbers less than 10: 2, 4, 6, 8
3. Write the final set in roster form: {2, 4, 6, 8}
7.
Are and the same set?
1. a)Cannot determine
2. b)Yes, order does not matter in sets
3. c)No, they are in different order
4. d)Only if we rearrange them
Answer: Yes, order does not matter in sets
Yes, they are the same set. In set notation, order does not matter. Both sets contain exactly the same elements: 1, 2, and 3.
8.
Simplify the set by removing duplicates. Write your answer using curly braces.
Answer: {1, 2, 3}
Sets contain only unique elements. Removing duplicates gives us .
9.
Convert to roster form:
Answer: {1, 2, 3, 4, 6, 12}
1. What is 12 divided by 1? 12
2. What is 12 divided by 2? 6
3. List all factors of 12: 1, 2, 3, 4, 6, 12
4. Write the set in roster form: {1, 2, 3, 4, 6, 12}
10.
Given , which statement is TRUE?
1. a)
2. b)
3. c)
4. d)
Answer:
is true because 9 is in the set. 5 and 15 are not in the set, so statements with them are false.
11.
Write a set-builder notation for the set .
Answer: {x | x is a multiple of 5, 1 <= x/5 <= 5}
1. What number do all elements have in common as a factor? 5
2. So these are all multiples of what number? 5
3. What is the smallest multiple? 5
4. What is the largest multiple? 25
12.
How many elements are in the set ?
Answer: 7
Primes less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19. The odd ones are: 3, 5, 7, 11, 13, 17, 19 (7 elements). We exclude 2 because it is even.
13.
Find all elements of and count them.
Answer: 9
1. What is the square root of 25? 5
2. So means is between which two values? -5 and 5
3. List all integers where : -4, -3, -2, -1, 0, 1, 2, 3, 4
4. How many elements are in the set? 9
14.
What is ?
1. a)
2. b) (empty set)
3. c)
4. d)
Answer: (empty set)
There are no integers between 0 and 1. The inequality excludes both 0 and 1, and there are no integers in between. So the set is empty: .
15.
Let and . Write a set-builder notation that describes set .
Answer: {x | x is odd, 1 <= x <= 9}
1. What do 1, 3, 5, 7, 9 have in common? odd
2. What is the smallest element? 1
3. What is the largest element? 9
4. Complete: odd
16.
Which of the following represents an empty set?
1. a)
2. b)
3. c)
4. d)
Answer:
is empty because nothing can be unequal to itself. contains zero. contains everything (infinite set).
17.
If , what is the cardinality of ? (Write just the number)
Answer: 5
The cardinality is the number of elements in set . Counting: a, b, c, d, e = 5 elements. So .
18.
Find and count the elements.
Answer: 7
1. What is ? 1
2. What is ? 49
3. What is ? 64
4. List all perfect squares from 1 to 50: 1, 4, 9, 16, 25, 36, 49
5. How many elements? 7