Introduction to Sets

Show your work for each problem.

1.
Which of the following is NOT a valid set?
1. a)
2. b)
3. c)
4. d)
Answer:
uses parentheses, which indicate an ordered tuple, not a set. Sets must use curly braces .
2.
Let . Which statement is TRUE?
1. a)
2. b)
3. c)
4. d)
Answer:
is true because 6 is one of the elements listed in . The others (3, 10, 1) are not in the set.
3.
How many elements are in the set ?
Answer: 5
The set contains 5 elements: a, e, i, o, and u. This is the cardinality of the set: .
4.
How many elements are in the empty set ?
Answer: 0
The empty set (also written as ) contains no elements at all. Its cardinality is 0: .
5.
Which two sets are equal?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
and are equal because sets ignore duplicates. simplifies to .
6.
If , what is the cardinality ?
Answer: 5
The cardinality is the number of elements in set . Counting 1, 3, 5, 7, 9, we get .
7.
What is the correct way to write "5 is not an element of set A"?
1. a)
2. b)
3. c)
4. d)
Answer:
correctly states that 5 is not an element of set A. The symbol is the negation of .
8.
Write the set of all even numbers between 1 and 10 (exclusive) in roster notation.
Answer: {2, 4, 6, 8}
1. What is the first even number greater than 1? 2
2. List all even numbers up to (but not including) 10. 2, 4, 6, 8
3. Write the set using curly braces. {2, 4, 6, 8}
9.
List all elements of the set .
Answer: 1, 2, 3, 4, 5
1. What is the smallest positive integer? 1
2. What is the largest positive integer less than 6? 5
3. List all positive integers from 1 to 5. 1, 2, 3, 4, 5
10.
Which statement about and is correct?
1. a)They are the same set
2. b) has 1 element, has 0
3. c) has 1 element, has 0
4. d)Both have 0 elements
Answer: has 1 element, has 0
is a set with one element (the number 0), so . The empty set has no elements, so .
11.
Given , determine which of 3, 6, and 9 are elements of .
Answer: 6
1. Is 3 in the set ? no
2. Is 6 in the set ? yes
3. Is 9 in the set ? no
4. Which number(s) are elements of E? 6
12.
If , how many elements does have?
Answer: 3
has 3 elements: 1, the set (counted as ONE element), and 4. So .
13.
What is ?
1. a)0
2. b)2
3. c)1
4. d)undefined
Answer: 1
is a set with exactly one element: the empty set . Therefore .
14.
Write the set of all prime numbers less than 15 in roster notation.
Answer: {2, 3, 5, 7, 11, 13}
1. What is the smallest prime number? 2
2. List all prime numbers less than 10. 2, 3, 5, 7
3. What prime numbers are between 10 and 15? 11, 13
4. Write the complete set. {2, 3, 5, 7, 11, 13}
15.
Let and . Are and equal? Explain.
Answer: yes
1. Does the order of elements matter in sets? no
2. What elements are in G? 1, 2, 3
3. What elements are in H? 1, 2, 3
4. Are G and H equal? yes
16.
Which is a correct set-builder notation for the set ?
1. a)
2. b)
3. c)
4. d)
Answer:
gives us even numbers greater than 1 and less than 9, which are exactly 2, 4, 6, 8.
17.
Write the set containing only the number 7 using roster notation. Use curly braces.
Answer: {7}
The set containing only 7 is written as . This is called a singleton set - a set with exactly one element.