The Induction Principle is of great importance in discrete mathematics: Number Theory, Graph Theory, Enumerative Combinatorics, Combinatorial Geometry, and other subjects.
Usually one proves the validity of a relationship f(n) = g(n) if one has a guess from small values of n.
Then one checks that f(1) = g(1), and, by making the assumption f(n) = g(n) for some n, one proves that also f(n + 1) = g(n + 1).
From this one concludes by the Induction Principle that f(n) = g(n) for all n ∈ N. There are many variations of this principle.
The relationship f(n) = g(n) is valid for 0 already, or, starting from some n0 > 1.
The inductive assumption is often f(k) = g(k) for all k < n, and, from this assumption, one proves the validity of f(n) = g(n).
We assume familiarity with all this and apply induction in unusual circumstances to make nontrivial proofs.
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Our goal at this course is to enhance our students’ mathematical intuition by focusing on a deep understanding of mathematical concepts and to enable them to link different concepts and apply their knowledge to solve mathematical problems to help them to improve their performance at Maths exams.
This course guides you through the fundamentals of Python programming using an interactive Python library known as Turtle.
This course encompasses a range of Geometry topics such as coordinate and spatial geometry, introductory trigonometry, angles, parallel lines, congruent and similar triangles, polygons, circles, the Pythagorean Theorem, and more. Emphasis will be placed on reinforcing Algebra skills and enhancing critical thinking through problem-solving in both mathematical and real-world contexts.