The midpoint of the line joining the points (a,b) and (c,d) is ((a+c)/2,(b+d)/2). The coefficient for this midpoint will have integer coefficients if and only if both a and c, and b and d have the same parity. In other words, they are both odd, or both even. Notice that the sum of two odd […]
Let’s start this exercise with two definitions: the definition of function and the pigeonhole principle. Definition: “Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element
Let’s start with the following definition. THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Source: Discrete Mathematics and its Applications by Rosen. Let’s have d boxes numbered 0,
Let’s start with the following definitions. THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” THE GENERALIZED PIGEONHOLE PRINCIPLE: “If N objects are placed into k boxes, then there
To solve this exercise, we will use the fact of how many reminders of the division there can be when dividing by a certain number. If you divide by n, you can get n possible reminders. These are 0, 1, 2, …, n-1. If we have a certain set of n consecutive integers, we can
Let’s revisit the two principles that apply to this type of exercise. “THE PIGEONHOLE PRINCIPLE If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” “THE GENERALIZED PIGEONHOLE PRINCIPLE If N objects are
Let’s start with the following definition. THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Discrete Mathematics and its applications by Rosen. The number of available letters in the
Let’s start with definitions. THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Discrete Mathematics and its applications by Rosen. Like in previous exercises, let’s find the boxes and
THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Discrete Mathematics and its applications by Rosen. Like in previous exercises, let’s follow the principle and define the boxes and
Let’s refresh the pigeonhole principle. THE PIGEONHOLE PRINCIPLE: “If k is a positive integer and k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.” Discrete Mathematics and its applications by Rosen. When you divide an integer number by 4,
