Problem 3 - Algebra and Trigonometry

Question

True or False The intersection of two sets is always a subset of their union.

Step-by-Step Solution

Answer

True

Step 1: Understand the key concepts

Review the definitions of 'intersection' and 'union' of sets. The intersection of two sets contains all elements that are common to both sets, while the union contains all elements from both sets.

Step 2: Define two sets

Consider two sets, A and B. For example, let A = {1, 2, 3} and B = {3, 4, 5}.

Step 3: Find the intersection

Determine the intersection of A and B, denoted as A ∩ B. In this case, A ∩ B = {3}, because 3 is the only element common to both sets.

Step 4: Find the union

Determine the union of A and B, denoted as A ∪ B. In this case, A ∪ B = {1, 2, 3, 4, 5}, which includes all elements from sets A and B.

Step 5: Check if intersection is a subset of union

By definition, set C is a subset of set D if every element of C is also an element in D. In this case, confirm that every element in A ∩ B is also in A ∪ B.

Step 6: Conclusion

Since the element '3' in A ∩ B is also in A ∪ B, we confirm that the intersection of two sets is always a subset of their union.

Key Concepts

Intersection of Sets

Understanding the concept of intersection is key in set theory. The intersection of two sets, denoted by \( A \cap B \), includes only the elements that are common to both sets.
For example, if we have sets A = {1, 2, 3} and B = {3, 4, 5}, their intersection \( A \cap B \) is {3}.
It’s important to note that if there are no common elements, the intersection is an empty set, denoted as \( \varnothing \).

• The intersection helps in understanding shared characteristics or elements.
• This concept is pivotal in mathematical proofs and applications.

Grasping this concept will allow you to tackle more complex problems in mathematics and logic.

Union of Sets

The union of sets, written as \( A \cup B \), combines all elements from both sets, eliminating any duplicates.
For instance, with sets A = {1, 2, 3} and B = {3, 4, 5}, their union \( A \cup B \) is {1, 2, 3, 4, 5}.
Every element from both sets is included in the union.
Key points to remember:

• The union encompasses all unique elements from the sets involved.
• Union operations are often useful in areas such as probability, logical operations, and database queries.

Understanding union operations adds to your overall comprehension of relationships between sets.

Subset

A set C is considered a subset of set D, denoted \( C \subseteq D \) if every element of C is also an element of D.
This means that C can be fully contained within D.
For example, let’s say C = {3} and D = {1, 2, 3, 4, 5}; clearly, every element of C is also in D.
This concept helps in defining hierarchical relationships among sets.
Important notes on subsets:

• If \( C \subseteq D \), there’s no element in C that isn’t in D.
• This is crucial in proofs, as subsets demonstrate containment and ordering within sets.

Knowing this, you can point out certain properties, like in our example where the intersection of two sets A and B is always a subset of their union.