Let sets A and B be defined as follows. A is the set of odd numbers greater than 7 and less than 23. B = {d, i, k, l, s, u, w, y}. (a) Find the cardinalities of A and B. n(A) = ? n... Let sets A and B be defined as follows. A is the set of odd numbers greater than 7 and less than 23. B = {d, i, k, l, s, u, w, y}. (a) Find the cardinalities of A and B. n(A) = ? n(B) = ? (b) Select true or false for: 17 \notin A, 21 \in A, d \in B, t \in B
Understand the Problem
The question defines two sets, A and B. Set A consists of odd numbers between 7 and 23, and set B consists of a set of letters. Part (a) asks for the cardinalities (number of elements) of sets A and B. Part (b) asks to determine whether the given elements are in their respective sets, requiring an understanding of set membership and the definitions of sets A and B.
Answer
$n(A) = 7$ $n(B) = 8$ $17 \notin A$ is False $21 \in A$ is True $d \in B$ is True $t \in B$ is False
Answer for screen readers
$n(A) = 7$ $n(B) = 8$
$17 \notin A$ is False $21 \in A$ is True $d \in B$ is True $t \in B$ is False
Steps to Solve
- Find the elements of set A
Set A contains odd numbers greater than 7 and less than 23. So, A = {9, 11, 13, 15, 17, 19, 21}.
- Find the cardinality of set A, n(A)
The cardinality of a set is the number of elements in the set. A = {9, 11, 13, 15, 17, 19, 21} has 7 elements. Therefore, $n(A) = 7$.
- Find the cardinality of set B, n(B)
Set B = {d, i, k, l, s, u, w, y} has 8 elements. Therefore, $n(B) = 8$.
- Determine if 17 is not in A
A = {9, 11, 13, 15, 17, 19, 21}. Since 17 is in A, the statement $17 \notin A$ is false.
- Determine if 21 is in A
A = {9, 11, 13, 15, 17, 19, 21}. Since 21 is in A, the statement $21 \in A$ is true.
- Determine if d is in B
B = {d, i, k, l, s, u, w, y}. Since d is in B, the statement $d \in B$ is true.
- Determine if t is in B
B = {d, i, k, l, s, u, w, y}. Since t is not in B, the statement $t \in B$ is false.
$n(A) = 7$ $n(B) = 8$
$17 \notin A$ is False $21 \in A$ is True $d \in B$ is True $t \in B$ is False
More Information
Cardinality refers to counting the number of elements in a set. The symbol $\in$ means "is an element of" and $\notin$ means "is not an element of".
Tips
A common mistake when finding the cardinality of set A is to forget the definition "greater than 7 and less than 23". This means not including 7 and 23 and only including the odd numbers in between.
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