This two-volume set collects and presents many fundamentals of mathematics in an enjoyable and elaborating fashion. The idea behind the two books is to provide substantials for assessing more modern developments in mathematics and to present impressions which indicate that mathematics is a fascinating subject with many ties between the diverse mathematical disciplines. The present volume examines many of the most important basic results in geometry and discrete mathematics, along with their proofs, and also their history. Contents Geometry and geometric ideas Isometries in Euclidean vector spaces and their classification in ℝ n The conic sections in the Euclidean plane Special groups of planar isometries Graph theory and platonic solids Linear fractional transformation and planar hyperbolic geometry Combinatorics and combinatorial problems Finite probability theory and Bayesian analysis Boolean lattices, Boolean algebras and Stone’s theorem Details.

Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Klyuch aktivacii nano antivirus. Blerina Xhabli. Discrete Mathematics Primes and Greatest Common Divisors 12/26.

The original implication is a little hard to analyze because there are so many different combinations of nine cards. But consider the contrapositive: If you don't have at least three cards all of the same suit, then you don't have nine cards.

It is easy to see why this is true: you can at most have two cards of each of the four suits, for a total of eight cards (or fewer). The converse: If you have at least three cards all of the same suit, then you have nine cards. This is false.

You could have three spades and nothing else. Note that to demonstrate that the converse (an implication) is false, we provided an example where the hypothesis is true (you do have three cards of the same suit), but where the conclusion is false (you do not have nine cards). Understanding converses and contrapositives can help understand implications and their truth values: Example 0.2.6 Suppose I tell Sue that if she gets a 93% on her final, then she will get an A in the class.

Assuming that what I said is true, what can you conclude in the following cases: • Sue gets a 93% on her final. • Sue gets an A in the class. • Sue does not get a 93% on her final. • Sue does not get an A in the class.

Of course there are many answers. It helps to assume that the statement is true and the converse is note true. Think about what that means in the real world and then start saying it in different ways. Some ideas: Use “necessary and sufficient” language, use “only if,” consider negations, use “or else” language.

7 Translate into symbols. Use (E(x) ) for “ (x ) is even” and (O(x) ) for “ (x ) is odd.” • No number is both even and odd. • One more than any even number is an odd number. • There is prime number that is even. • Between any two numbers there is a third number. • There is no number between a number and one more than that number.

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