b) Determine whether the function F from $$ \{a, b, d\} $$ to $$ \{1,2,3,4,5\} $$ with $$ f(a)=4, f(b)=5, f(c)=1 $$ and $$ f(d)=3 $$ is one to one. c) Given that $$ A=\{1,3,5\} $$ and $$ B=\{2,4,6\} $$ find i. $$ A \cap B $$ ii. $$ A \cup B $$

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### Part b: Determine if the function $$ F $$ is one-to-one

A function is considered one-to-one (or injective) if different inputs map to different outputs. In other words, if $$ f(x_1) = f(x_2) $$ implies that $$ x_1 = x_2 $$.

Given the function $$ F $$ defined as follows:
- $$ f(a) = 4 $$
- $$ f(b) = 5 $$
- $$ f(c) = 1 $$ (Note: $$ c $$ is not in the domain $$ \{a, b, d\} $$)
- $$ f(d) = 3 $$

We can see the outputs for the inputs in the domain:
- $$ f(a) = 4 $$
- $$ f(b) = 5 $$
- $$ f(d) = 3 $$

The outputs $$ 4, 5, $$ and $$ 3 $$ are all distinct. Since there are no two different inputs that map to the same output, the function $$ F $$ is one-to-one.

### Part c: Set operations on $$ A $$ and $$ B $$

Given the sets:
- $$ A = \{1, 3, 5\} $$
- $$ B = \{2, 4, 6\} $$

i. **Find $$ A \cap B $$** (the intersection of $$ A $$ and $$ B $$):
The intersection of two sets includes only the elements that are present in both sets. Since there are no common elements between $$ A $$ and $$ B $$:
$$
A \cap B = \emptyset
$$

ii. **Find $$ A \cup B $$** (the union of $$ A $$ and $$ B $$):
The union of two sets includes all elements that are in either set. Thus:
$$
A \cup B = \{1, 3, 5\} \cup \{2, 4, 6\} = \{1, 2, 3, 4, 5, 6\}
$$

### Summary of Results
- The function $$ F $$ is one-to-one.
- $$ A \cap B = \emptyset $$
- $$ A \cup B = \{1, 2, 3, 4, 5, 6\} $$
The function $$ F $$ is one-to-one. The intersection of $$ A $$ and $$ B $$ is the empty set, and the union of $$ A $$ and $$ B $$ is $$ \{1, 2, 3, 4, 5, 6\} $$.

b) Determine whether the function F from to with
and is one to one.
c) Given that and
find
i.
ii.

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b) Determine whether the function F from to with and is one to one. c) Given that and find i. ii.