Discrete Math - 8.5.1 The Principle of Inclusion-Exclusion

Summary

This video explores the principle of inclusion-exclusion (PIE) through practical examples, including real estate and number theory. It begins with a scenario involving homes for sale, demonstrating how to calculate total distinct homes using Venn diagrams and PIE. The video then applies PIE to find the number of integers divisible by specific numbers and explains how to extend this principle to three or more sets. By providing a general formula, it emphasizes the effectiveness of PIE in simplifying complex counting problems without the need for visual aids, making it a valuable tool in combinatorial mathematics.

Takeaways

• 😀 The principle of inclusion-exclusion is a method for counting the size of the union of multiple sets without double-counting elements.
• 🏠 In a real estate example, there are 64 homes with garages, 21 with swimming pools, and 17 with both features, totaling 68 unique homes.
• 📊 A Venn diagram can visually represent the intersections between sets, but inclusion-exclusion allows for a quicker mathematical calculation.
• ➕ For two sets A and B, the formula for finding the union is |A ∪ B| = |A| + |B| - |A ∩ B|.
• 📏 The floor function is useful for counting integers within a range that are divisible by specific numbers, such as 7 and 11.
• 💻 The inclusion-exclusion principle can also be applied to real-world scenarios like counting college majors to avoid double-counting students enrolled in multiple disciplines.
• 🔢 For three sets, the formula expands to include subtracting the intersections of each pair and adding back the intersection of all three sets.
• 🔄 The generalization of inclusion-exclusion allows for the calculation of unions for any number of sets, with alternating signs for intersections.
• 📉 This counting technique is essential in various fields such as mathematics, computer science, and probability.
• 🧩 Understanding and applying the principle of inclusion-exclusion can simplify complex counting problems without the need for visual aids.

Q & A

What is the Principle of Inclusion-Exclusion (PIE)?

The Principle of Inclusion-Exclusion is a counting technique used to calculate the total number of elements in the union of multiple sets, accounting for overlaps to avoid double counting.

In the real estate example, how many total homes were calculated to be for sale?

A total of 68 homes were calculated to be for sale, considering the overlaps of homes with garages and pools.

How is the total number of homes determined using a Venn diagram?

The total is found by adding the number of homes with garages only, homes with pools only, and homes with both features, avoiding double counting.

What formula is used in PIE to find the union of two sets?

The formula is |A ∪ B| = |A| + |B| - |A ∩ B|, where |A| and |B| are the sizes of sets A and B, respectively, and |A ∩ B| is the size of their intersection.

What is the significance of the intersection in PIE?

The intersection accounts for elements that belong to both sets; it must be subtracted to prevent counting these elements twice.

In the example of positive integers not exceeding 1000, how many were found to be divisible by 7 or 11?

220 positive integers not exceeding 1000 were found to be divisible by either 7 or 11.

What are the steps to extend PIE to three sets?

To extend to three sets, add the sizes of each set, subtract the sizes of all pairwise intersections, and then add back the size of the intersection of all three sets.

What did the video illustrate with the example of permutations of digits?

The video demonstrated how to use PIE to find permutations of digits under specific conditions involving their positions.

What is the generalized formula for PIE with 'n' sets?

The generalized formula is |A1 ∪ A2 ∪ ... ∪ An| = ∑ |Ai| - ∑ |Ai ∩ Aj| + ∑ |Ai ∩ Aj ∩ Ak| - ... + (-1)^(n+1) |A1 ∩ A2 ∩ ... ∩ An|.

Why is the principle of inclusion-exclusion considered efficient?

PIE allows for quick calculations of unions without needing to draw diagrams, providing a clear formulaic approach to combinatorial counting.