Question Statement

Let $A=\{2,3,4,5\}$ and $B=\{b,c,d,e\}$. A function $f:A\to B$ is defined by the set of ordered pairs: $f=\{(2,b),(3,c),(4,e),(5,e)\}$

Determine whether the function $f$ is one-to-one, into, or onto.

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Background and Explanation

To solve this, we must understand the definitions of function mappings. A function is one-to-one (injective) if every element in the domain maps to a unique element in the codomain. It is onto (surjective) if the range equals the codomain, and into if the range is a proper subset of the codomain (i.e., there is at least one element in the codomain not mapped to by the domain).

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Solution

To determine the nature of the function, we analyze the mapping of each element from domain $A$ to codomain $B$.

1. Checking if the function is One-to-One

A function is one-to-one if $f(x_1)=f(x_2)$ implies $x_1=x_2$. From the given set of ordered pairs:

• $f(2)=b$
• $f(3)=c$
• $f(4)=e$
• $f(5)=e$

Since $f(4)=e$ and $f(5)=e$, we have two different inputs ($4$ and $5$) mapping to the same output ($e$). $f(4)=f(5)=e\text{ but }4\ne 5$ Therefore, the function is not one-to-one (it is a many-to-one function).

2. Checking if the function is Into or Onto

To decide if a function is onto or into, we compare the Range of the function with the Codomain $B$.

• The Codomain $B$ is given as $\{b,c,d,e\}$.
• The Range (the set of actual outputs) is $\{b,c,e\}$.

We can see that the element $d\in B$ does not have a corresponding pre-image in set $A$. In mathematical terms: $\text{Range }f=\{b,c,e\}\subset B$ Since the Range is a proper subset of the Codomain ($\text{Range}\ne \text{Codomain}$), the function is into.

Conclusion: The function is not one-to-one and it is into.

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Key Formulas or Methods Used

• One-to-one (Injective): $f(a)=f(b)⟹a=b$ for all $a,b\in A$.
• Onto (Surjective): $\text{Range}(f)=\text{Codomain}(B)$.
• Into: $\text{Range}(f)\subset \text{Codomain}(B)$ and $\text{Range}(f)\ne \text{Codomain}(B)$.

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Summary of Steps

1. List the outputs for every input in the domain $A$.
2. Identify if any two different inputs result in the same output (to check for one-to-one status).
3. Identify the Range of the function by listing all unique second elements in the ordered pairs.
4. Compare the Range to the Codomain $B$. If they are not equal, the function is "into."