Problem 39 - Discrete Mathematics with Applications

Question

A recent survey by the MAD corporation indicates that of the 700 families interviewed, 220 own a television set but no stereo, 200 own a stereo but no camera, 170 own a camera but no television set, 80 own a television set and a stereo but no camera, 80 own a stereo and a camera but no television set, 70 own a camera and a television set but no stereo, and 50 do not have any of these. Find the number of families with: Exactly two of the items.

Step-by-Step Solution

Answer

There are 130 families owning exactly two of the items.

Step 1: Assign variables to the given information

Let A = families owning TV only, B = families owning stereo only, C = families owning a camera only, AB = families owning television and stereo but not a camera, BC = families owning a stereo and a camera but not a television set, and AC = families owning a camera and a television set but not a stereo. We have the following variable assignments: A = 220 B = 200 C = 170 AB = 80 AC = 70 BC = 80 We also know that 50 families have none of these appliances, and that a total of 700 families were interviewed.

Step 2: Calculate total families owning at least one item

There are 700 families, out of which 50 own none of these items. Hence, there are 700 - 50 = 650 families owning at least one item.

Step 3: Apply the Principle of Inclusion and Exclusion (PIE)

The total families owning exactly two items can be found using PIE as follows: \( Number\_of\_Two\_Items = (A + B + C) - 2*(AB + AC + BC) + 3*Intersection\_of\_all\_Three \) Note that no families own all three items (not given in the problem statement), so Intersection_of_all_Three = 0. Now, substituting the values from Step 1: \( Number\_of\_Two\_Items = (220 + 200 + 170) - 2*(80 + 70 + 80) \)

Step 4: Compute the families with exactly two items

We can now simplify and compute the number of families owning exactly two items: \( Number\_of\_Two\_Items = (590) - 2*(230) = 590 - 460 = 130 \) So there are 130 families owning exactly two of the items.

Key Concepts

Set Theory

Set theory is the branch of mathematical logic that studies collections of objects, which are referred to as sets. Objects within these sets are called elements or members. At its core, set theory is about understanding relationships between different groups and their elements. In our exercise, each family represents an element, and their ownership of different items (television set, stereo, camera) represents different sets.

For instance, set A (families owning TVs only) is distinct from set B (families owning stereos only), but there might be families that belong to both sets, like those who own TVs and stereos but not cameras (set AB). Here, we apply the basic ideas of set theory to organize our approach to the problems, defining clear groups and understanding how they can intersect.

Venn Diagrams

Venn diagrams are visual representations of sets and the relationships between different sets, marked by overlapping circles. Each circle represents a set, with overlaps showing shared elements. They are fantastic tools in comprehending problems in set theory and combinatorics.

In the context of our problem, we could draw three intersecting circles for the families owning TVs, stereos, and cameras. Where two circles overlap, it represents families owning exactly two of the items. The space where all three overlap would represent families that own all three, which in this case, is zero as per the given data. Through Venn diagrams, the scenario is not just abstract numbers but becomes a clear, visual representation of how these sets combine and relate, aiding in problem-solving.

Combinatorics

Combinatorics is the area of mathematics focused on counting, combination, and permutation of various elements within a set. It's all about figuring out how many different ways things can be arranged or grouped. In our exercise, combinatorics comes into play as we calculate the number of families with exactly two of the items, without double-counting any family.

The Principle of Inclusion-Exclusion (PIE) is a key concept in combinatorics that helps solve complex counting problems like the one presented. By adding up the individual counts and then subtracting the overlaps (families counted in multiple sets), we effectively avoid over-counting. This approach is essential in combinatorics to ensure accurate results in enumeration tasks.