Sure, let's address each case and find the range of the relation defined on the set [tex]\( \{1, 2, 3, 4\} \)[/tex]:
### a. "a square of"
For each element in our set [tex]\( \{1, 2, 3, 4\} \)[/tex], we need to find its square:
- [tex]\( 1^2 = 1 \)[/tex]
- [tex]\( 2^2 = 4 \)[/tex]
- [tex]\( 3^2 = 9 \)[/tex]
- [tex]\( 4^2 = 16 \)[/tex]
Thus, the range of the relation "a square of" is [tex]\( \{1, 4, 9, 16\} \)[/tex].
### b. "a square root of"
For each element in our set [tex]\( \{1, 2, 3, 4\} \)[/tex], we need to find its square root:
- [tex]\( \sqrt{1} = 1.0 \)[/tex]
- [tex]\( \sqrt{2} \approx 1.4142135623730951 \)[/tex]
- [tex]\( \sqrt{3} \approx 1.7320508075688772 \)[/tex]
- [tex]\( \sqrt{4} = 2.0 \)[/tex]
Thus, the range of the relation "a square root of" is [tex]\( \{1.0, 1.4142135623730951, 1.7320508075688772, 2.0\} \)[/tex].
### c. "equal"
Since this relation implies that each element maps to itself, the range will simply be the elements of our set:
[tex]\[ \{1, 2, 3, 4\} \][/tex]
### d. "half of"
For each element in our set [tex]\( \{1, 2, 3, 4\} \)[/tex], we need to find half of it:
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( \frac{2}{2} = 1.0 \)[/tex]
- [tex]\( \frac{3}{2} = 1.5 \)[/tex]
- [tex]\( \frac{4}{2} = 2.0 \)[/tex]
Thus, the range of the relation "half of" is [tex]\( \{0.5, 1.0, 1.5, 2.0\} \)[/tex].
### e. "cube of"
For each element in our set [tex]\( \{1, 2, 3, 4\} \)[/tex], we need to find its cube:
- [tex]\( 1^3 = 1 \)[/tex]
- [tex]\( 2^3 = 8 \)[/tex]
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 4^3 = 64 \)[/tex]
Thus, the range of the relation "cube of" is [tex]\( \{1, 8, 27, 64\} \)[/tex].
### f. "cube root of"
For each element in our set [tex]\( \{1, 2, 3, 4\} \)[/tex], we need to find its cube root:
- [tex]\( \sqrt[3]{1} = 1.0 \)[/tex]
- [tex]\( \sqrt[3]{2} \approx 1.2599210498948732 \)[/tex]
- [tex]\( \sqrt[3]{3} \approx 1.4422495703074083 \)[/tex]
- [tex]\( \sqrt[3]{4} \approx 1.5874010519681994 \)[/tex]
Thus, the range of the relation "cube root of" is [tex]\( \{1.0, 1.2599210498948732, 1.4422495703074083, 1.5874010519681994\} \)[/tex].
Summarizing, the ranges are:
- a. "a square of": [tex]\( \{1, 4, 9, 16\} \)[/tex]
- b. "a square root of": [tex]\( \{1.0, 1.4142135623730951, 1.7320508075688772, 2.0\} \)[/tex]
- c. "equal": [tex]\( \{1, 2, 3, 4\} \)[/tex]
- d. "half of": [tex]\( \{0.5, 1.0, 1.5, 2.0\} \)[/tex]
- e. "cube of": [tex]\( \{1, 8, 27, 64\} \)[/tex]
- f. "cube root of": [tex]\( \{1.0, 1.2599210498948732, 1.4422495703074083, 1.5874010519681994\} \)[/tex]
