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In mathematics, a set is simply a collection of distinct objects, typically referred to as elements. For instance, in the exercise, we have two sets with elements listed within curly brackets: \(\{3, 6, 9, 12\}\) and \(\{5, 10, 15\}\). Here, the numbers 3, 6, 9, and 12 are elements of the first set, and 5, 10, and 15 are elements of the second set.
Sets can consist of anything such as numbers, letters, or even other sets. It’s important to note that order does not matter in sets; for example, \(\{3, 6, 9, 12\}\) is the same as \(\{12, 9, 6, 3\}\). Additionally, elements are unique within a set, meaning a set cannot have duplicate elements.

The intersection of sets is a fundamental concept in set theory and is denoted by the \(\cap\) symbol. It refers to a new set that contains all the elements that are present in both of the original sets. Think of it as the 'overlap' or 'commonality' between the sets. Consider the exercise at hand: \(\{3, 6, 9, 12\} \cap \{5, 10, 15\}\). To find the intersection, you compare the two sets to see which elements, if any, are present in both. In this example, you notice there are no elements that appear in both sets:

• The first set: \(\{3, 6, 9, 12\}\)
• The second set: \(\{5, 10, 15\}\)

Since there are no shared elements, the intersection is an empty set.

An empty set, also known as a null set, is a set that does not contain any elements. In set notation, it’s represented as \(\emptyset\) or \(\{\}\). The empty set is a unique set in mathematics because it is the only set with no elements.
In our exercise, we discovered that the intersection of \(\{3, 6, 9, 12\}\) and \(\{5, 10, 15\}\) resulted in the empty set because there were no common elements. This tells us that there is no overlap between the two sets, illustrating a key concept in understanding set intersections.
Recognizing an empty set is crucial in many areas of mathematics since it often forms the basis for further operations or proofs within set theory and other mathematical disciplines.