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.

Jan 29, 2011 - of my all-time favorite mathematical threads into the general narrative. Option themes of IB: geometry, discrete mathematics, abstract alge- bra, series and ordinary. CHAPTER 1 Advanced Euclidean Geometry. Discrete mathematics is the study of mathematical structures that are fundamentally discrete. In mathematics, they are useful in geometry and certain parts of topology, e.g. Biggs (2002-12-19).

• 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.

Like this are among the objects studied by discrete mathematics, for their interesting, their usefulness as models of real-world problems, and their importance in developing computer. Discrete mathematics is the study of that are fundamentally rather than. In contrast to that have the property of varying 'smoothly', the objects studied in discrete mathematics – such as,, and in – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in 'continuous mathematics' such as.

Discrete objects can often be by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with (finite sets or sets with the same as the natural numbers). However, there is no exact definition of the term 'discrete mathematics.' Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of, such as,,,,.

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Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

In university curricula, 'Discrete Mathematics' appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by and into a course that is basically intended to develop in freshmen; therefore it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike in this respect. The is awarded for outstanding papers in discrete mathematics. Much research in was motivated by attempts to prove that all maps, like this one, can be using so that no areas of the same color share an edge.