Math Floor Proof

However today we want try another approach that works well here and in other important cases where a contrapositive proof may not.

Math floor proof. For any integer m and any real number x if x is not an integer then floor x floor m x m 1. Given any x r the ceiling of x denoted dxe is the least integer greater than than x. The floor function also known as the entier function is defined as having its value the largest integer which does not exceed its argument. The floor and ceiling of the number are the integers to the immediate left and to the immediate right of the number unless the number is itself an integer in which case its floor and ceiling both equal the number itself.

In mathematics and computer science the floor function is the function that takes as input a real number and gives as output the greatest integer less than or equal to denoted or similarly the ceiling function maps to the least integer greater than or equal to denoted or. We introduce the floor and ceiling functions then do a proof with them. Floor and ceiling imagine a real number sitting on a number line. Given any x r the floor of x denoted bxc is the greatest integer less than x.

Assume n is odd and show that n3 5 is even. The second approach works well for this problem. Mat231 transition to higher math proof by contradiction fall 2014 3 12. Like and share the video if it helped.

X r n z bxc n n x n 1. I m confused as to how that s even true. Direct proof and counterexample v. A contrapositive proof seems more reasonable.

When applied to any positive argument it represents the integer part of the argument obtained by suppressing the fractional part. Acoustics is a branch of physics that deals with the study of mechanical waves in gases liquids and solids including topics such as vibration sound ultrasound and infrasound a scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an acoustical engineer the application of acoustics is present in almost all. Discrete math 3 5 floor and ceiling p. Danziger 1 floor and ceiling definition 1 1.

For example if x and m are the same number it doesn t work i don t think.