Pure Mathematics A1: Sets, functions and vectors

44 flashcards · Mathematics · by lakesare

  1. What's the origin of the adjective Cartesian?

    From the surname of René Descartes (published this idea in 1637). René Descartes The adjective Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.

  2. What's the famous book by Euclid?

    Elements.

  3. How to read this notation?

    mod k

  4. What's a plane figure?

    Plane figure is a set of points in R2. Plane figure is also known as plane set. The lines and circles are simple examples of plane sets.

  5. How would you specify this plane set?

  6. How to denote: A is a subset of B VS A is a proper subset of B?

    A is a subset of B - ⊆ (either proper subset or =!) A is a proper subset of B - ⊂

  7. How would you show that A is a proper subset of B (A ⊂ B)?

    The point (0, 2) lies in B, but not in A. We should simply show A and B are not equal (it's a matter of 1 element!)

  8. Can it be so that both A ⊆ B and A ⊂ B?

    Very much so, A ⊆ B means "either A ⊂ B or A = B".

  9. Simplify this expression.

  10. Simplify this expression.

  11. What does x ↦ f (x) denote?

    The rule of the function.

  12. What is the role of the codomain?

    It is convenient to know where the output lies. Notice the codomain must not be restrictive - it should always be at least equal to the range.

  13. What's a transformation function? Or simply transformation.

    Transformations are functions that have a geometric interpretation (e.g. f: R2 → R2).

  14. What's the rule of the identity function iA?

  15. What's meant by the "image of a set S under a function f"?

    f(S)

  16. Find f({ 1, 2, 3 }). Image of a set S under a function f.

  17. Why is the idea of the image of a set useful in geometry?

    We frequently want to consider the result of the transformation on a plane figure.

  18. What does f(domain_set) denote?

    f(domain_set) = range_set

  19. Name 1 step to determine if the function is one-to-one.

    1. We algebraically show that if 2 elements of the domain have equal images under f, then they must be the same element.

  20. To have an inverse function f –1, does f need to be: one-to-one, onto?

    one-to-one - f needs to be one-to-one to have f-1. onto - not necessary, but it creates a cute situation where f and f-1 are inverses of each other.

  21. What's the domain of the inverse function (f –1)?

    f –1: f(A) → A

  22. What does this notation denote?

    A unit vector.

  23. What's the alternative notation for i, j, k unit vectors?

  24. What is the position vector of the point P(2, -1)?

    OP = 2i - j

  25. How to find AB? AB is a vector from A to B.

  26. What do we want when we say we want a vector form of the equation of the line?

    We want r(λ), where r is a position vector of every point on the line.

  27. How would we find r(λ)? (Given p, q position vectors, find r)

  28. How is the angle between 2 vectors defined?

    θ ∈ [o, π]

  29. How is the dot product (between vectors u and v) defined in this Pure Mathematics module?

  30. What is the equation of the (x, y)-plane?

    {(x, y, z) ∈ ℝ3 | z = 0 }

  31. Imagine all normals to some plane.

    All of these are normals.

  32. Imagine all planes that some vector is normal to.

    Any vector n in ℝ3 is a normal vector to infinitely many planes, all parallel to each other.

  33. How would we find the distance formula for ℝ3? The distance formula for ℝ2 is found via the Pythogorean theorem.

  34. How would you find the vector equation of the plane, given: the normal n = (a, b, c) and the point in the plane P(x1, x2, x3). Any vector n in ℝ3 is a normal vector to infinitely many planes, all parallel to each other.

    We have: PX ⋅ n = 0 (because n is a normal). PX = x – p (x – p) ⋅ n = 0 x ⋅ n = p ⋅ n

  35. What to do when: you want to find the vector equation you want to find the usual equation of some figure?

    vector equation - aim for the position vector of any particular point in your figure. usual equation - aim for the (x, y, z) point of any particular point in your figure.

  36. How would you go about showing the equality of these sets?

    Algebraically, with reasoning and drawings! Show A ⊆ B Show B ⊆ A

  37. Name 2 steps to determine if the function is onto.

    Find the range of the function Guess the range of the function. Show that our guessed range = codomain (each set is a subset of another set). Determine if range = codomain.

  38. Name 2 steps to find the range of the function.

    Guess the range of the function. Show that our guessed range = codomain (each set is a subset of another set).

  39. Name 2 steps to find the inverse of the function.

    Find the range of f (this will be the domain of f-1!). Find f-1.

  40. When we say that the function is "restricted" - what exactly is restricted?

    Function's domain.

  41. What does ℝ∗ denote?

    The set of all real numbers, except 0.

  42. What 2 functions can be composed together (g ◦ f)?

    Any 2 functions. We just might need to restrict the domain of f. This definition a llows us to consider the c omposite of any two functions, though in some cases t he domain may turn out to be the empty set ∅.However, some texts insist on f(A) ⊆ C as a condition to ensure g ◦ f exists.

  43. What's iA VS iB?

    iA maps A → A, iB maps B → B.

  44. How would you prove that function g is the inverse of function f?

    Show g(f(x)) = x Show f(g(x)) = x