91Ó°ÊÓ

In set theory, the union of two sets is a fundamental concept that combines the elements of both. When you see an expression like \(A \cup B\), it involves creating a new set that contains all unique elements found in either set \(A\) or set \(B\).
Specifically, this means you take every element from set \(A\), every element from set \(B\), and put them all together while keeping duplicates out.

• If \(x\) is an element of \(A\), or \(x\) is an element of \(B\), or both, then \(x\) is in \(A \cup B\).

Now, considering the situation where \(A \subseteq B\), this means every element of \(A\) is already hiding inside \(B\). Therefore, when you unite them, all elements of \(A\) are already accounted for within \(B\). Thus, the union \(A \cup B\) is simply the set \(B\) itself. This sometimes simplifies to just saying the bigger or covering set, \(B\), represents the union without needing to add any elements from \(A\).

The intersection of sets, denoted by \(A \cap B\), is about finding common ground between two sets, \(A\) and \(B\). If you picture this with a Venn diagram, the intersection is often represented by the overlapping area.
Essentially, you're looking for elements that exist in both sets simultaneously.

• If an element \(x\) is in set \(A\) and that same element is in set \(B\), then \(x\) belongs to the intersection \(A \cap B\).

When one set is a subset of another, as in \(A \subseteq B\), it simply means every element of \(A\) is found within \(B\). This results in the intersection \(A \cap B\) logically simplifying to just set \(A\), as all elements of \(A\) are present in \(B\). So, in this configuration, the intersection highlights the fact that set \(A\) itself lies within \(B\).

The concept of set difference, denoted as \(A \backslash B\), describes the elements that uniquely belong to set \(A\), and not in set \(B\). It's a way to explore what makes two sets distinct.
If you're subtracting the elements of \(B\) from \(A\), you are left with those in \(A\) that are not shared with \(B\).

• Mathematically, \(A \backslash B\) contains all elements \(x\) such that \(x\) is in \(A\) but not in \(B\).

Now, if \(A \subseteq B\), every element of \(A\) is also present in \(B\). This means there are no elements in \(A\) that aren't already included in \(B\). Consequently, \(A \backslash B\) results in an empty set, often represented by \(\emptyset\), indicating there are no remaining elements exclusive to \(A\). In this scenario, the set difference emphasizes the complete inclusion of set \(A\) within set \(B\).