Question

In a class of 35 students who are either biology majors or have blonde hair, there are 27 biology majors and 21 blondes. How many biology majors must be blonde?

Short answer

13 biology majors must be blonde.

Step-by-step solution

Define Given Variables

Let's define the variables from the problem: Let \( B \) be the set of biology majors and \( H \) be the set of students with blonde hair. \( |B| = 27 \) and \( |H| = 21 \). The total is \( |B \cup H| = 35 \).

Apply the Set Union Formula

The formula for the union of two sets is \( |B \cup H| = |B| + |H| - |B \cap H| \). We need to find \( |B \cap H| \), which represents biology majors with blonde hair.

Plug in the Known Values

Substitute the known values into the union formula: \( 35 = 27 + 21 - |B \cap H| \).

Solve for the Intersection

Rearrange the equation to find \( |B \cap H| \):\[|B \cap H| = 27 + 21 - 35\]\[|B \cap H| = 48 - 35\]\[|B \cap H| = 13\]

Conclusion

Therefore, 13 biology majors must have blonde hair.

Key concepts

Intersection of Sets

The concept of the intersection of sets is quite intuitive once we break it down. When you hear "intersection," think of common elements shared between two or more sets. In this case, the term "intersection" refers to anything that appears simultaneously in both sets. For instance, if we have Set B representing biology majors and Set H representing students with blonde hair, the intersection \( B \cap H \) will consist of students who are biology majors and also have blonde hair.

Using our example, if we know the total number of biology majors \(|B|\) and the total number of blonde-haired students \(|H|\), finding the intersection tells us how many belong to both categories. This is useful in various scenarios where you need to identify shared characteristics or features.

Understanding the intersection is crucial when calculating overlaps, as seen in many fields of mathematics and real-world applications like market research and data analytics. Remember, the intersection of sets \(A\) and \(B\) is denoted as \(A \cap B\) and consists of all elements that \(A\) and \(B\) agree upon.

Union of Sets

The union of sets is a fundamental operation in set theory that represents the total distinct elements found in one set, the other, or both. Imagine compiling items from all involved sets into one grand set—this is your union.

When analyzing two sets, say Set B and Set H as in our example, the union \( B \cup H \) includes all elements present in either B or H. It's essential to remember, though, that in the context of union, any element that appears in both sets is only counted once. This helps avoid duplication, ensuring our union reflects a comprehensive collection of unique elements.

The set union formula is \( |B \cup H| = |B| + |H| - |B \cap H| \). It starts by summing the individual sizes, then subtracts the intersection size to remove duplicate counts of shared elements. In the exercise, this formula is critical to solving how many biology majors must have blonde hair after accounting for those who appear in both Set B and Set H.

Set Operations

Set operations are tools that allow us to perform specific actions on sets, much like how arithmetic operations work on numbers. They help us understand and solve various problems involving collections of items, such as the number of students who belong to certain categories.

Common set operations include:

• Union: Denoted by \( \cup \), combines all unique elements of involved sets.
• Intersection: Denoted by \( \cap \), identifies shared elements across sets.
• Difference: Finds elements in one set not appearing in the other, denoted as \( A - B \) (or \( A \setminus B \)).
• Complement: Concerns elements not in the set, taking the wider "universal set" into account.

In our exercise, understanding these operations helps in calculating the number of biology majors with blonde hair by clarifying steps through the procedural application of union and intersection formulas. Mastery of set operations is not just academic—it's practical for solving real-world problems where categorization and overlap occur.