real numbers, intervals, and inequalities
... In the following discussion we will be concerned with certain sets of real numbers, so it will be helpful to review the basic ideas about sets. Recall that a set is a collection of objects, called elements or members of the set. In this text we will be concerned primarily with sets whose members are ...
... In the following discussion we will be concerned with certain sets of real numbers, so it will be helpful to review the basic ideas about sets. Recall that a set is a collection of objects, called elements or members of the set. In this text we will be concerned primarily with sets whose members are ...
Maximum Product Over Partitions Into Distinct Parts
... 1 as a part. It follows from the relation p · 1 < p + 1, valid for all positive integers p. Another observation, p + q < pq, valid for all 2 ≤ p < q, implies that a longer partition is preferred over a shorter one. Hence, the product of parts will be maximized by long partitions that do not contain ...
... 1 as a part. It follows from the relation p · 1 < p + 1, valid for all positive integers p. Another observation, p + q < pq, valid for all 2 ≤ p < q, implies that a longer partition is preferred over a shorter one. Hence, the product of parts will be maximized by long partitions that do not contain ...
2.3 Infinite sets and cardinality
... As well as understanding this example at the informal/intuitive level suggested by the picture above, think about the formula above, and satisfy yourself that it does indeed descibe a bijection between N and Z. If you are convinced that the two questions above (and all others like them) have unique ...
... As well as understanding this example at the informal/intuitive level suggested by the picture above, think about the formula above, and satisfy yourself that it does indeed descibe a bijection between N and Z. If you are convinced that the two questions above (and all others like them) have unique ...
1. Complex Numbers and the Complex Exponential
... reasons that will become increasingly clear as the course progresses, we will use the language of complex numbers and we will insist that the students become both comfortable and familiar with this language. Please note that we are not asking for the knowledge contained in a complex analysis course ...
... reasons that will become increasingly clear as the course progresses, we will use the language of complex numbers and we will insist that the students become both comfortable and familiar with this language. Please note that we are not asking for the knowledge contained in a complex analysis course ...
Chapter 8 Number Theory
... Eg. Any subset of size six from S={1,2,3,4,5,6,7,8,9} must contain two elements whose sum is 10. (Sol.) The numbers: 1,2,3,4,5,6,7,8,9 are pigeons. {1,9}, {2,8}, {3,7}, {4,6}, {5} are pigeonholes. When 6 pigeons go to their respective pigeons, they must fill at least one of the two-element subsets w ...
... Eg. Any subset of size six from S={1,2,3,4,5,6,7,8,9} must contain two elements whose sum is 10. (Sol.) The numbers: 1,2,3,4,5,6,7,8,9 are pigeons. {1,9}, {2,8}, {3,7}, {4,6}, {5} are pigeonholes. When 6 pigeons go to their respective pigeons, they must fill at least one of the two-element subsets w ...
introduction to proofs - Joshua
... The Principle of Mathematical Induction is that completing both steps proves that the statement is true for all natural numbers greater than or equal to the initial number i . For the example statement about odd numbers and squares, the intuition behind the principle is first that the base step dire ...
... The Principle of Mathematical Induction is that completing both steps proves that the statement is true for all natural numbers greater than or equal to the initial number i . For the example statement about odd numbers and squares, the intuition behind the principle is first that the base step dire ...
B4 Identifying and represetning positive integers on a number line
... a number line is a picture of a straight line on which every point corresponds to a real number and every real number to a point. ...
... a number line is a picture of a straight line on which every point corresponds to a real number and every real number to a point. ...
