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17. Module No#08
This chapter delves into the principles of discrete mathematics, specifically focusing on the application of the pigeonhole principle to prove the existence of certain properties among sets of integers or points in a plane. The fundamental aim is to showcase how pigeonhole logic can help derive relationships, affirm conditions, and establish the validity of mathematical statements across different scenarios.
Sections
This section covers key concepts in discrete mathematics using the pigeonhole principle to prove the existence of certain properties among integers and points in a coordinate system.
This section discusses the application of the pigeonhole principle in proving mathematical statements related to midpoint calculations of integer coordinates and the existence of integer pairs whose sum equals a specific number.
This lecture explores the application of the pigeonhole principle in proving mathematical statements related to integer coordinates and specific sums.
This section discusses the application of the pigeonhole principle in proving the existence of certain mathematical properties involving distinct points in two-dimensional planes and integers.
This section discusses the application of the pigeonhole principle to prove the existence of a pair of distinct points in a two-dimensional plane with integer coordinates whose midpoint also has integer coordinates.
In this section, the pigeonhole principle is applied to show that among any 5 integers chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}, there always exists at least one pair of integers whose sum is 9.
In this section, the pigeonhole principle is used to demonstrate the existence of multiples of an integer that consist solely of the digits 0 and 1 in decimal form.
The pigeonhole principle can be utilized to prove mathematical assertions involving distinct integers or points.
Irrespective of the arbitrary selection of integers, certain pairs will always possess defined relationships, such as summing to a specific value.
Mathematical proofs can demonstrate the availability of multiples of integers with specific digit representations.
Pigeonhole Principle
A principle that states if n items are put into m containers with n > m, then at least one container must contain more than one item.
Midpoint Formula
The formula used to find the midpoint of a line segment defined by two endpoints (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2).
Decimal Expansion of Numbers
A representation of numbers in the base-10 numeral system, consisting of digits 0 through 9 placed in specific positional values.
Practice Exercises
Total Questions
2
Estimated Time
4 min
Passing Score
70%
Instructions
- Read each question carefully
- You can use hints if you need help
- Complete all questions before submitting
1 more question available
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