Question

Given that \(\Phi: \mathrm{A} \rightarrow \mathrm{B}\) is a function with a set \(\mathrm{C}\) containing the image of \(\phi\) prove that \(\Phi: \mathrm{A} \rightarrow \mathrm{C}\) is also a function.

Short answer

To prove that \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) is a function, we must show that it associates every element from \(\mathrm{A}\) to exactly one element from \(\mathrm{C}\). Since \(\Phi : \mathrm{A} \rightarrow \mathrm{B}\) is a function and \(\mathrm{C}\) contains the image of \(\Phi\), for every \(x \in \mathrm{A}\), there exists a unique \(y \in \mathrm{C}\) such that \(\Phi (x) = y\). Therefore, \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) satisfies the definition of a function.

Step-by-step solution

Recall the definition of a function

A function is a relation between two sets that associates each element from the first set to exactly one element from the second set. If we can show that \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) has this property, then we can conclude that it is indeed a function.

Show that \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) associates each element from \(\mathrm{A}\) to exactly one element from \(\mathrm{C}\)

Let's consider an arbitrary element \(x \in \mathrm{A}\). Since \(\Phi : \mathrm{A} \rightarrow \mathrm{B}\) is a function, by definition, it associates \(x\) with exactly one element from \(\mathrm{B}\). We can denote this as \[ \Phi (x) = y \in \mathrm{B}, \] where \(y\) is uniquely determined by \(x\). Now, by the given information, we know that \(\mathrm{C}\) contains the image of \(\Phi\). This means that if \(\Phi (x) = y\), then \(y \in \mathrm{C}\). Therefore, for the same \(x \in \mathrm{A}\), we can write \[ \Phi (x) = y \in \mathrm{C}. \] Since \(x\) was chosen arbitrarily, it follows that every element from \(\mathrm{A}\) is associated with exactly one element from \(\mathrm{C}\) through \(\Phi\).

Conclusion

We have shown that \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) associates each element from \(\mathrm{A}\) to exactly one element from \(\mathrm{C}\). Thus, according to the definition of a function, we can conclude that \(\Phi : \mathrm{A} \rightarrow \mathrm{C}\) is indeed a function.

Key concepts

Definition of a Function

In mathematics, a function is a special kind of relation that uniquely associates each element from one set, called the **domain**, to exactly one element in another set, called the **codomain**. This association is often represented by a rule or formula. If we take sets \(\mathrm{A}\) and \(\mathrm{B}\), a function \(\Phi\) is a mapping from elements of \(\mathrm{A}\) to elements of \(\mathrm{B}\) such that every element \(x\) in \(\mathrm{A}\) is paired with a unique element \(y\) in \(\mathrm{B}\).
Consider the function \(\Phi: \mathrm{A} \rightarrow \mathrm{B}\). For each \(x \in \mathrm{A}\), there exists a unique \(y \in \mathrm{B}\) such that \(\Phi(x) = y\). This is the core characteristic that distinguishes functions from generic relationships: the uniqueness and definiteness of the pairing.

• A function is defined by exactly pairing each element of the domain with a single element of the codomain.
• A function can be represented with different notations like \(f(x), g(x),\) or in this case \(\Phi(x)\).

Image of a Function

The image of a function refers to the set of all outcomes we get by applying the function to every element of the domain set. In our context, it is the collection of values in the codomain that the function actually maps to. This means that it contains every element \(y\) for which there exists at least one input \(x\) such that \(\Phi(x) = y\).
For the function \(\Phi: \mathrm{A} \rightarrow \mathrm{B}\), the image is made up of elements in \(\mathrm{B}\) that are mapped from \(\mathrm{A}\). The image is essentially a snapshot of what the function can produce from its domain.

• If the image of \(\Phi\) is a subset of \(\mathrm{C}\), this means all outcomes of the function \(\Phi\) will lie within \(\mathrm{C}\).
• The image set can sometimes be a proper subset of the entire codomain.

Knowing and identifying the image is crucial, especially when discussing the mappings between sets.

Mapping Between Sets

Mapping is the process of assigning each element from a set, known as the **domain**, to an element in another set, called the **codomain**. In functional terms, these sets are \(\mathrm{A}\) and \(\mathrm{B}\) respectively. A mapping must assign, or "map," each element \(x\) from the domain \(\mathrm{A}\) to exactly one element \(y\) in the codomain \(\mathrm{B}\).
The idea of mapping is related to drawing arrows from each element in \(\mathrm{A}\) pointing to an element in \(\mathrm{B}\). The mapping \(\Phi: \mathrm{A} \rightarrow \mathrm{C}\) confirms that this one-to-one association holds even when the elements of \(\mathrm{A}\) are mapped into a subset of \(\mathrm{B}\), which is \(\mathrm{C}\) here, the image set.

• A proper mapping ensures every element of the domain is paired with a single, specific element of the codomain.
• Mapping can show relationships and transformations from one mathematical space to another.

Accurately understanding the mapping strategy between sets helps in analyzing function-related problems better.