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In set theory, a subset is a set in which all the elements are also contained in another set. This relationship is symbolized mathematically as \(A \subseteq B\), where \(A\) is a subset of \(B\).
This means that every element in \(A\) must also be an element of \(B\). Whenever you're considering whether one set is a subset of another, ask yourself: 'Is every single item in my first set also present in the second one?'
Keep in mind:

• If \(A\) and \(B\) contain exactly the same elements, \(A\) is still a subset of \(B\).
• An empty set, represented as \(\emptyset\), is always considered a subset of any given set.

Visualizing subsets can help. Imagine set \(A\) as a smaller box fitting entirely into the larger box of set \(B\).
In the example given, the set \(\{V, C, R\}\) is contained within the set \(\{V, C, R, S\}\), ensuring that \(\{V, C, R\} \subseteq \{V, C, R, S\}\).

While subsets simply mean one set's elements are entirely found in another, proper subsets take this a step further. A proper subset, denoted as \(A \subset B\), signifies that \(A\) is not only a subset of \(B\), but \(B\) also has at least one element that \(A\) doesn't have.This additional condition ensures that \(A\) is not equal to \(B\), and \(A\) indeed "properly" sits inside \(B\). Proper Subset Quick Checklist:

• All elements of \(A\) are in \(B\).
• \(B\) has extra elements not found in \(A\).

In our example, set \(B = \{V, C, R, S\}\) includes all elements of set \(A = \{V, C, R\}\), plus one extra element, \(S\). Therefore, \(A\) is a proper subset of \(B\).
This means \(A \subset B\) is true.

Mathematical reasoning is critical when working with concepts like subsets and proper subsets. It involves understanding definitions, applying logic, and carefully comparing elements in sets.Using mathematical reasoning, you:

• Identify conditions under which a statement is true or false.
• Verify each condition systematically, as seen in the step-by-step solution.
• Apply definitions (like for subsets and proper subsets) to specific examples.

This process strengthens your problem-solving skills, allowing you to make accurate conclusions.
When reasoning through subsets, test each element: is it contained in the larger set? For proper subsets, examine if the larger set has extra elements.
This reasoning confirms that in the exercise, \(A \subseteq B\) is true because all elements match, and \(A \subset B\) is also true, thanks to the additional element in \(B\). Keep practicing this method to boost your mathematical skills!