Functions Defined on General Sets

1
Functions Defined on General Sets

• Lecture 30
• Section 7.1
• Wed, Apr 5, 2006

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Relations

• A relation R from a set A to a set B is a subset
  of A ? B.
• If x ? A and y ? B, then x has the relation R to
  y if (x, y) ? R.
• We may also write x R y.

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Examples Relations

• Let A B R. Let x, y ? R. Define x R y to
  mean that y x2.
• Describe R.
• Let A B R. Let x, y ? R. Define x R y to
  mean that y lt x2.
• Describe R.
• Is R ? R a relation?
• Is ? a relation?

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Functions

• Let A and B be sets.
• A function from A to B is a relation from A to B
  with the property that for every x ? A, there
  exists exactly one y ? B such that (x, y) ? f.
• Write f A ? B and f(x) y.
• A is the domain of f.
• B is the co-domain of f.

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Functions

• Note that functions and algebraic expressions are
  two different things.
• For example, do not confuse the algebraic
  expression (x 1)2 with the function
  f R ? R defined by f(x) (x 1)2.

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Examples Functions

• f R ? R by f(x) x2.
• g R ? R ? R by g(x, y) 1 x y.
• h R ? R ? R ? R by h(x, y) (-x, -y).
• For any set A, k ?(A) ? ?(A) ? ?(A) by k(X, Y)
  X ? Y.
• For any sets A and B, m ?(A) ? ?(B) by m(X) X
  ? B.

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Examples Functions

• Let n be the size of a complete binary tree.
  Define f N ? R by f(n) the average number of
  nodes visited to locate a randomly selected value
  in the tree.
• We found earlier that if the tree contains 2n 1
  nodes, then

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Inverse Images

• If f(x) y, we say that y is the image of x and
  that x is an inverse image of y.
• The inverse image of y is the set
• f -1(y) x ? X f(x) y.
• In the previous examples, find
• f -1(4).
• g-1(0).
• h-1(5, 10).

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Equality of Functions

• Let f X ? Y and g X ? Y be two functions.
• Then f g if f(x) g(x) for all x ? X.
• Are the functions f(x) x and g(x) ??x2
  equal?
• Are the functions f(x) 1 and g(x) sec2 x
  tan2 x equal?

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Boolean Functions

• A Boolean function is a function whose domain is
  0, 1 ? ? 0, 1 (or 0, 1n) and codomain is
  0, 1.
• Example Let p, q be Boolean variables and
  define f(p, q) p ? q.

p q f(p, q)
1 1 1
1 0 0
0 1 0
0 0 0
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The Number of Boolean Functions

• How many Boolean functions are there in 2
  variables?
• What are they?
• How many Boolean functions are there in 3
  variables?
• How many Boolean functions are there in n
  variables?

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Boolean Functions

• What Boolean function is defined by
• f(x, y) xy?
• What Boolean function is defined by
• f(x, y) x y xy?
• What Boolean function is defined by
• f(x) 1 x?
• What Boolean function is defined by
• f(x, y, z) 1 xy z xyz?