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Summary

  • Ordered pair: A pair of elements grouped together in a particular order.
  • Cartesian product: A × B of two sets A and B is given by A × B = {(a, b): a A, b B}
    In particular R × R = {(x, y): x,
    y R} and R × R × R = (x, y, z): x, y, z R}
  • If (a, b) = (x, y), then a = x and b = y.
  • If n(A) = p and n(B) = q, then n(A × B) = pq.
  • A × φ = f
  • In general, A × B B × A.
  • Relation: A relation R from a set A to a set B is a subset of the Cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B.
  • The image of an element x under a relation R is given by y, where (x, y) R.
  • The domain of R is the set of all first elements of the ordered pairs in a relation R.
  • The range of the relation R is the set of all second elements of the ordered pairs in a relation R.
  • Function: A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write f: A B, where f(x) = y.
  • A is the domain and B is the codomain of f.
  • The range of the function is the set of images.
  • A real function has the set of real numbers or one of its subsets both as its domain and as its range.
  • Algebra of functions: For functions f : X R and g : X R, we have
    (f + g) (x) = f(x) + g(x),
    x X.

    (f - g) (x) = f (x) - g(x), x X.

    (f.g) (x) = f (x) .g (x), x X.

    (kf) (x) = k f (x) ), x X.





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