More Functions and Sets

1
More Functions and Sets

• Rosen 1.8

2
Inverse Image

• Let f be an invertible function from set A to set
  B. Let S be a subset of B. We define the
  inverse image of S to be the subset of A
  containing all pre-images of all elements of S.
• f-1(S) a?A f(a) ?S

3
Let f be an invertible function from A to B. Let
S be a subset of B. Show that f-1(S) f-1(S)

• What do we know?
• f must be 1-to-1 and onto

4
Let f be an invertible function from A to B. Let
S be a subset of B. Show that f-1(S) f-1(S)

• Proof We must show that f-1(S) ? f-1(S) and
  that f-1(S) ? f-1(S) .
• Let x ? f-1(S). Then x?A and f(x) ? S. Since
  f(x) ? S, x ? f-1(S). Therefore x ? f-1(S).
• Now let x ? f-1(S). Then x ? f-1(S) which
  implies that f(x) ? S. Therefore f(x) ? S and x
  ? f-1(S)

5
Let f be an invertible function from A to B. Let
S be a subset of B. Show that f-1(S) f-1(S)

• Proof
• f-1(S) x?A f(x) ? S Set builder notation
• x?A f(x) ? S Def of Complement
• f-1(S) Def of Complement

6
Floor and Ceiling Functions

• The floor function assigns to the real number x
  the largest integer that is less than or equal to
  x. ?x?
• ?x? n iff n ? x lt n1, n?Z
• ?x? n iff x-1 lt n ? x, n?Z
• The ceiling function assigns to the real number x
  the smallest integer that is greater than or
  equal to x. ?x?
• ?x? n iff n-1 lt x ? n, n?Z
• ?x? n iff x ? n lt x1, n?Z

7
Examples

• ?0.5? 1
• ?0.5? 0
• ?-0.3? 0
• ?-0.3? -1
• ?6? 6
• ?6? 6
• ?-3.4? -3
• ?3.9? 3

8
Prove that ?xm? ?x? m when m is an integer.

• Proof Assume that ?x? n, n?Z.
• Therefore n ? x lt n1.
• Next we add m to each term in the inequality to
  get nm ? xm lt nm1.
• Therefore ?xm? nm ?x? m

?x? n iff n ? x lt n1, n?Z
9
Let x?R. Show that ?2x? ?x? ?x1/2?

• Proof Let n?Z such that ?x? n. Therefore
• n ? x lt n1. We will look at the two cases
• x ? n 1/2 and x lt n 1/2.
• Case 1 x ? n 1/2
• Then 2n1 ? 2x lt 2n2, so ?2x? 2n1
• Also n1 ? x 1/2 lt n2, so ?x 1/2 ? n1
• ?2x? 2n1 n n1 ?x? ?x1/2?

10
Let x?R. Show that ?2x? ?x? ?x1/2?

• Case 2 x lt n 1/2
• Then 2n ? 2x lt 2n1, so ?2x? 2n
• Also n ? x 1/2 lt n1, so ?x 1/2 ? n
• ?2x? 2n n n ?x? ?x1/2?

11
Characteristic Function

• Let S be a subset of a universal set U. The
  characteristic function fS of S is the function
  from U to 0,1such that fS(x) 1 if x?S and
  fS(x) 0 if x?S.
• Example Let U Z and S 2,4,6,8.
• fS(4) 1
• fS(10) 0

12
Let A and B be sets. Show that for all x,
fA?B(x) fA(x)fB(x)

• Proof fA?B(x) must equal either 0 or 1.
• Suppose that fA?B(x) 1. Then x must be in the
  intersection of A and B. Since x? A?B, then x?A
  and x?B. Since x?A, fA(x)1 and since x?B fB(x)
  1. Therefore fA?B fA(x)fB(x) 1.
• If fA?B(x) 0. Then x ? A?B. Since x is not in
  the intersection of A and B, either x?A or x?B or
  x is not in either A or B. If x?A, then fA(x)0.
  If x?B, then fB(x) 0. In either case fA?B
  fA(x)fB(x) 0.

13
Let A and B be sets. Show that for all x,
fA?B(x) fA(x) fB(x) - fA(x)fB(x)

• Proof fA?B(x) must equal either 0 or 1.
• Suppose that fA?B(x) 1. Then x?A or x?B or x is
  in both A and B. If x is in one set but not the
  other, then fA(x) fB(x) - fA(x)fB(x)
  10(1)(0) 1. If x is in both A and B, then
  fA(x) fB(x) - fA(x)fB(x) 11 (1)(1) 1.
• If fA?B(x) 0. Then x?A and x?B. Then fA(x)
  fB(x) - fA(x)fB(x) 0 0 (0)(0) 0.

14
Let A and B be sets. Show that for all x,
fA?B(x) fA(x) fB(x) - fA(x)fB(x)

• A B A?B fA?B(x) fA(x) fB(x) - fA(x)fB(x)
• 1 1 1 1 11-(1)(1) 1
• 1 0 1 1 10-(1)(0) 1
• 0 1 1 1 01-(0)(1) 1
• 0 0 0 0 0)-(0)(0) 0