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49 results in How to Think Like a Mathematician

Page 3 of 3

  • 23 - Techniques of proof III: Contradiction

  • from IV - Techniques of proof
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 161-165

  • Summary

    Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.

    Ralph Waldo Emerson, Journal entry, 8 November 1838

    The law of the excluded middle asserts that a statement is true or it is false, it cannot be anything in between. We can use this as another method of proof. We assume that the statement is false and proceed logically to show that this gives a statement that we definitely know is false such as 1 = 0 or the Moon is made of cheese. Thus our assumption must be wrong, the statement can't be false – it leads to something ridiculous – so the statement is true.

    This method is called proof by contradiction. The name comes from the fact that assuming that the statement is false is later contradicted by some other fact. It is also known by the name reductio ad absurdum which when translated means reduction to the absurd.

    Simple examples of proof by contradiction

    The first example is just to show you the idea of proof by contradiction. The statement is easier to prove by a direct method as we have seen in Theorem 20.1.

    Example 23.1

    Suppose that n is an odd integer. Then n2 is an odd integer.

    Proof. Assume the contrary. That is, we suppose that n is an odd integer but that the conclusion is false, i.e. n2 is an even integer.

  • 17 - Proof

  • from III - Definitions, theorems and proofs
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 116-118

  • Summary

    Do not confuse reasons which sound good with good, sound reasons.

    Anon.

    Mathematics is fantastic. It is a subject where we do not have to take anyone's word or opinion. The truth is not determined by a higher authority who says ‘because I say so’, or because they saw it in a dream, the pixies at the bottom of their garden told them, or it came from some ancient mystical tradition. The truth is determined and justified with a mathematical proof.

    What is a proof?

    A proof is an explanation of why a statement is true. More properly it is a convincing explanation of why the statement is true. By convincing I mean that it is convincing to a mathematician. (What that means is an important philosophical point which I am not going to get into; my interest is more in practical matters.)

    Statements are usually proved by starting with some obvious statements, and proceeding by using small logical steps and applying definitions, axioms and previously established statements until the required statement results.

    The mathematician's concept of proof is different to everyday usage. In everyday usage or in court for instance, proof is evidence that something is likely to be true. Mathematicians require more than this. We like to be 100% confident that a statement has been proved. We do not like to be ‘almost certain’.

  • 24 - Techniques of proof IV: Induction

  • from IV - Techniques of proof
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 166-174

  • Summary

    One thing leads to another.

    Anon.

    Induction is a very powerful technique used regularly by mathematicians. Initially, it can be confusing because it looks like we assume what is to be proved. As we know, that never proves theorems. On the plus side, spotting when to use it is easy and we need only check two conditions to apply it.

    Induction is applied when we have an infinite number of statements indexed by the natural numbers such as

    ‘n 5 – n even for all n ∈ ℕ’.

    It is not sufficient to prove this for a sample of natural numbers, whether that sample involves hundreds, millions or even billions of numbers; we have to prove it for all n.

    With induction we don't prove the statements directly. What we do is perhaps best described by analogy with domino toppling. As you are probably aware, this is where dominoes are standing on their ends in such a way that when you push the first one over, it knocks the second domino over, that in turn knocks the third down, and so on. Provided the dominoes are arranged so that each knocks down the next, then all of them will fall.

    The process of induction is that we prove that ‘if the kth statement is true, then the k + 1th statement is true’, i.e. the truth of one statement implies the truth of the next one. This is analogous to one domino knocking down the next one.

  • 30 - Injective, surjective, bijective – and a bit about infinity

  • from V - Mathematics that all good mathematicians need
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 218-229

  • Summary

    Listen, there are no measurements in infinity. You humans have got such limited little minds.

    The Doctor in Doctor Who and the Masque of Mandragora

    We have seen that sets are building blocks of mathematics and have said a little about functions between sets. We shall now look more closely at functions. For functions f : X → Y we define injective, surjective and bijective functions. These definitions allow us to compare sets and in the case of bijective functions allow us to say whether one set is just a relabelling of the elements of the other.

    Furthermore, using the notion of a bijection we can define two different types of infinite sets, those for which we can count the elements, such as ℕ, and those for which we can't count, such as ℝ. Thus we have two types of infinity!

    Injective functions

    Definition 30.1

    A function f : X → Y is called injective or one-to-one if, for all x1 ∈ X, x2 ∈ X, x1 ≠ x2 implies that f(x1) ≠ f(x2).

    The definition says that if I take two elements of X, then their values under f are the same if and only if the elements are the same. What we do not want is, for example, f(3) = f(5).

  • 9 - Converse and equivalence

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 75-79

  • Summary

    But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton [steamboat inventor], they laughed at the Wright brothers. But they also laughed at Bozo the Clown.

    Carl Sagan, Broca's Brain, 1979

    Statements of the form A ⇒ B are at the heart of mathematics. We have seen that for an implication A ⇒ B we can take its inverse (not(A) ⇒ not(B)) and its contrapositive (not(B) ⇒ not(A)). In this chapter we will look at another implication: B ⇒ A; this is called the converse of A ⇒ B. We shall see that a statement and its converse are not the same. One may be true and the other false, both may be true or both may be false.

    If A ⇒ B and B ⇒ A are both true, then we say that A and B are equivalent statements. Mathematicians really like equivalent statements, particularly if the A and B seem to have no obvious connection.

    The converse

    Definition 9.1

    The converse of the statement ‘A ⇒ B’ is ‘B ⇒ A’.

    The converse of

    ‘If I am Winston Churchill, then I am English’

    is

    ‘If I am English, then I am Winston Churchill.’

    This simple example shows that, even if a particular statement is true, its converse need not true. It may be true or it may not be true. Investigation is required before we can say.

  • 10 - Quantifiers – For all and There exists

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 80-83

  • Summary

    You can fool all the people some of the time, and some of the people all the time, but you cannot fool all the people all the time.

    Abraham Lincoln

    In Chapter 6, Making a statement, we called a sentence like ‘x is an odd number’ a conditional statement. Whether or not it was true depended on x. In this chapter we shall describe two fundamental ways of quantifying x so that we get statements rather than conditional statements.

    We can exemplify them via the following. The sentence ‘x2 = 2’ is conditional on x. By combining such sentences with some description of the x we can form statements. For example, ‘For all x ∈ ℤ, x2 = 2’ or ‘There exists an x ∈ ℝ such that x2 = 2.’ Note that the latter is true and the former is false.

    Since we are giving a description of how many of x we are talking about, i.e. assigning a quantity, we call these descriptions quantifiers. The fancy titles for the quantifiers we are interested in are universal quantifier and existential quantifier.

    For all – the universal quantifier

    Definition 10.1

    The phrase ‘for all’ is the universal quantifier. It is denoted ∀. (This is an upside down A. Think of the A in ‘All’ to remember this.)

  • 14 - Definitions, theorems and proofs

  • from III - Definitions, theorems and proofs
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 99-102

  • Summary

    The highest form of pure thought is in mathematics.

    Plato

    We now come to the heart of how mathematicians organize and present their work. Look through any high-level mathematics book (or see Chapter 1) and you will find that mathematics is not presented as one continuous piece of prose like a novel. Instead the text is divided up into small nuggets of information such as a Theorem, Proposition, Lemma, Corollary, Proof, Definition and Conjecture. All of these have special meanings and we will see how to approach them in the following chapters.

    Meanings

    We shall now briefly describe the meanings of the above words.

  • Definition: an explanation of the mathematical meaning of a word.

  • Theorem: a very important true statement.

  • Proposition: a less important but nonetheless interesting true statement.

  • Lemma: a true statement used in proving other true statements.

  • Corollary: a true statement that is a simple deduction from a theorem or proposition.

  • Proof: the explanation of why a statement is true.

  • Conjecture: a statement believed to be true, but for which we have no proof.

  • Axiom: a basic assumption about a mathematical situation.

    • Definitions

    Much more will be said about definitions in the next chapter. For the moment let us say that in higher mathematics close attention must be paid to definitions – much more than at lower levels. Definitions allow us to separate one class of objects from another or single out some interesting property.

  • 5 - How to solve problems

  • from I - Study skills for mathematicians
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 41-50

  • Summary

    It isn't that they can't see the solution. It is that they can't see the problem.

    G.K. Chesterton, The Scandal of Father Brown – ‘The Point of a Pin’

    Solving mathematical problems is hard and there is no magic formula which solves all problems. If there were, anyone could do it, and employers would not be so keen to hire problem-solving mathematicians. However, in this chapter we will try to provide at least some ideas on how to solve problems.

    Let us first distinguish between an exercise and a problem. An exercise is something that can be solved by a routine method, for example finding the roots of a quadratic. A problem is something that will require more thought, for example, we have to draw together a number of ideas and perhaps apply a number of the routine methods learned through exercises, such as root finding, in a new combination to get the answer. However, in this book, to save writing out the phrase ‘exercises and problems’, we will group problems in with exercises. Nonetheless, knowing that there is really a distinction can help.

    Much lower-level mathematics involves selecting and applying the right technique or formula to answer a question. While this is also true for some higher-level mathematics, we have the task of verifying the truth of statements, e.g. ‘There is an infinite number of prime numbers.’ The techniques needed for this type of problem are different from those of an apply-the-right-technique exercise.

Page 3 of 3