Question

Let \(F\) be a function, and let \(C, D \subseteq \operatorname{domain}(F)\). a.Prove that range \((F|C\cap D)\) \(\subseteq\) range \((F \mid c) \cap\) range \((F \mid D)\). (b) Show by example that equality need not hold in part (a).

Short answer

The range of a function over \(C \cap D\) is a subset of the intersection of individual ranges, but they are not necessarily equal.

Step-by-step solution

Understand the Problem

We need to prove that the range of the function restricted to the intersection of sets \(C\) and \(D\) is a subset of the intersection of the ranges of the function restricted to \(C\) and \(D\). Moreover, we must provide an example demonstrating that this subset relationship can be strict (i.e., the ranges need not be equal).

Analyze Range Behavior

When \(F\) is restricted to \(C \cap D\), its range includes only those outputs that correspond to inputs belonging solely to this subset. Thus, we denote this range as \(\text{range}(F|C \cap D)\). Similarly, \(\text{range}(F|C)\) includes outputs when inputs are in \(C\) and \(\text{range}(F|D)\) includes outputs when inputs are in \(D\).

Prove Subset Relationship

Consider any value \(y\) in \(\text{range}(F|C \cap D)\). Then there exists an \(x \in C \cap D\) such that \(F(x) = y\). Since \(x \in C\), \(y\) must also be in \(\text{range}(F|C)\), and since \(x \in D\), \(y\) must be in \(\text{range}(F|D)\). Therefore, \(y \in \text{range}(F|C) \cap \text{range}(F|D)\), proving the subset relationship: \(\text{range}(F|C\cap D) \subseteq \text{range}(F|C) \cap \text{range}(F|D)\).

Provide Counter-Example for Equality

Consider \(F(x) = x^2\) over \(\mathbb{R}\) with \(C = [-2, 0]\) and \(D = [0, 2]\). Here, \(C \cap D = \{0\}\) since it is the only point common to both intervals.\(\text{range}(F|C \cap D) = \{0\}\), but \(\text{range}(F|C) = [0, 4]\) and \(\text{range}(F|D) = [0,4]\). Their intersection is \([0,4]\), so \(\{0\} \subset [0,4]\) and equality does not hold here.

Key concepts

range of a function

The range of a function represents all possible output values that the function can produce based on its input values. When we say "range", we are usually referring to the set of all outputs for a particular function. To better understand this, imagine a machine where you input different values (like oranges, apples) and it gives you juice (outputs). The range is like all the different types of juices you can make with what you put in.
In mathematical terms, if you have a function \(F\) and it takes inputs from a set called the domain, the range will be all the outputs that you can get from putting every possible input from the domain into \(F\). Each function may act differently depending on its domain, like limiting specific types of inputs can change what you get out (its range). When a function \(F\) is restricted to a subset, like \(C \cap D\) from the domain, it affects the range. It only considers the output where inputs are strictly from \(C \cap D\). This influences the result because fewer inputs mean you might get fewer types of outputs.

subset relationship

A subset relationship helps us understand how different sets relate to each other. When we say that set \(A\) is a subset of set \(B\), we mean that every element of \(A\) is also an element of \(B\). Notationally, we write this as \(A \subseteq B\). In the context of functions, like in the exercise, we use subsets to talk about ranges. When \(F|C \cap D\) talks about a function restricted to the intersection of subsets \(C\) and \(D\), it means the range for these inputs is actually a subset of the ranges when inputs are from \(C\) and the range when inputs are from \(D\) separately. This subset idea helps validate certain mathematical arguments or prove if something is fully enclosed in another thing. Just like saying all believers in magic are believers in the world, showing that \( \text{range}(F|C \cap D) \subseteq \text{range}(F|C) \cap \text{range}(F|D) \) offers insight into function behavior.

counter-example

When trying to prove or disprove a statement, providing a counter-example is an effective approach. A counter-example is a specific case where the general statement fails or does not hold. By showing just one instance where the claim is wrong, it invalidates the claim or demonstrates its limitations.
For instance, in the exercise, we are asked to show that although \( \text{range}(F|C \cap D) \subseteq \text{range}(F|C) \cap \text{range}(F|D) \), equality does not always hold. To do this, consider \( F(x) = x^2 \) with specific sets like \( C = [-2, 0] \) and \( D = [0, 2] \). Here, the common part (the intersection) occurs only at \(0\), making its range just \(\{0\}\). Meanwhile, checking ranges individually gives us bigger outputs.
Through such cases, the counter-example clarifies why sometimes the sets don't fully match, helping to cement understanding of subset relationships in mathematics.