To find the domain and range of the relation [tex]\( S = \{ (2, a), (0, 2), (3, c), (3, b) \} \)[/tex], let's understand the definitions first:
1. Domain: The domain of a relation is the set of all first elements (or inputs) in the ordered pairs of the relation.
2. Range: The range of a relation is the set of all second elements (or outputs) in the ordered pairs of the relation.
Now, let's identify the domain and range from the given set [tex]\( S \)[/tex].
Step-by-Step Solution:
1. List the ordered pairs in the relation [tex]\( S \)[/tex]:
[tex]\[
S = \{ (2, a), (0, 2), (3, c), (3, b) \}
\][/tex]
2. Identify the first elements (those will form the domain):
- The first element of the pair [tex]\( (2, a) \)[/tex] is [tex]\( 2 \)[/tex].
- The first element of the pair [tex]\( (0, 2) \)[/tex] is [tex]\( 0 \)[/tex].
- The first element of the pair [tex]\( (3, c) \)[/tex] is [tex]\( 3 \)[/tex].
- The first element of the pair [tex]\( (3, b) \)[/tex] is [tex]\( 3 \)[/tex].
3. Combine these first elements into a set (since sets do not include duplicates):
[tex]\[
\text{Domain} = \{ 0, 2, 3 \}
\][/tex]
4. Identify the second elements (those will form the range):
- The second element of the pair [tex]\( (2, a) \)[/tex] is [tex]\( a \)[/tex].
- The second element of the pair [tex]\( (0, 2) \)[/tex] is [tex]\( 2 \)[/tex].
- The second element of the pair [tex]\( (3, c) \)[/tex] is [tex]\( c \)[/tex].
- The second element of the pair [tex]\( (3, b) \)[/tex] is [tex]\( b \)[/tex].
5. Combine these second elements into a set (again, no duplicates):
[tex]\[
\text{Range} = \{ a, 2, c, b \}
\][/tex]
So, the domain and range of the relation [tex]\( S \)[/tex] are:
[tex]\[
\text{Domain} = \{ 0, 2, 3 \}
\][/tex]
[tex]\[
\text{Range} = \{ a, 2, c, b \}
\][/tex]
