Functions and relations

1
Section 4.1

• Functions and relations

2
Examples of functions from algebra or pre-calculus

• f R ? R with rule f(x) 3x 1
• g Z ? N with rule g(x) x2
• Components of a function are the name, the domain
  (the set of inputs), the codomain (set which
  includes all outputs), and the rule that
  associates to each input one and only one output.
  
• Note that not every element of the codomain is
  necessarily an output, but every element of the
  domain must be a valid input.

3
Other ways to describe a functions rule

• Let A 1, 2, 3, 4, 5, B 1, 2, 3, 4, 5
• The following are equivalent descriptions of the
  rule for a function f A ? B
• The set of ordered pairs (1,5), (2,2), (3,1),
  (4,2), (5,5)
• The table
• The arrow diagram on the next slide

x 1 2 3 4 5
f(x) 5 2 1 2 5
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Other ways to describe a functions rule

• Let A 1, 2, 3, 4, 5, B 1, 2, 3, 4, 5.
  The following is the arrow diagram for the
  function whose rule is (1,5), (2,2), (3,1),
  (4,2), (5,5)

1 2 3 4 5
1 2 3 4 5
A B
5
Practice

• Let A x, y, z. Draw the arrow diagram for the
  function f P(A) ? P(A) with rule f (S) S x

P(A) P(A)
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Practice

• Let A x, y, z and B 0, 1, 2, 3, 4, 5.
  Draw the arrow diagram for the function f P(A)
  ? B with rule f (S) n(S)

0 1 2 3 4 5
P(A) B
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Binary relations

• A binary relation with domain A and codomain B is
  simply any subset of A B. These arise when we
  want to capture a general relationship between
  objects in A and objects in B.
• Example
• If A N and B N, then the relation R (a,
  b) ? A B a lt b captures the less than
  relationship among natural numbers. That is, R
  (0,1), (0,2), (1,2), (0,3), (1,3), (2,3),

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Examples of binary relations

• If A current SU students and B current SU
  courses, then the relation R with the rule,
  (x, y) ? R if person x takes course y, captures
  basic scheduling information.
• If A all film actors ever, then the relation
  R with the rule, (a, b) ? R if person x was in a
  film with person y, is essential in playing the
  Kevin Bacon game.
• We usually call this latter example a relation
  on A instead of a relation from A to A.

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Arrow diagrams for relations

• Draw the arrow diagram for the following relation
  R on A 1, 2, 3, 4, 5 with the rule, (m,n) ? R
  if m n is divisible by 3.

1 2 3 4 5
1 2 3 4 5
A A
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One-set arrow diagrams

• Draw the one-set arrow diagram for the relation R
  on A P(1, 2, 3) with the rule, (X,Y) ? R if
  X ? Y. Whats missing in the solution below?

2
1
3

1,2,3
2,3
1,2
1,3
A
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Practice

• Complete the one-set arrow diagram for the
  relation R on A P(1, 2, 3) with the rule,
  (X,Y) ? R if X ? Y ?.

2
1
3

1,2,3
2,3
1,2
1,3
A
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When is a binary relation a function?

• The additional condition for being a function is
  every input has one and only one output. How
  does this manifest itself under the different
  representations of a relation?

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When is a binary relation a function?

• In each case, one of these is a function from A
  to B and one is not.
• Example 1. Let A 1, 2, 3, 4, 5 and B 1, 2,
  , 9, 10.
• The relation with the rule (x,y) ? A B y
  x2
• The relation with the rule (x,y) ? A B y
  2x
• Example 2. Let A 1, 2, 3, 4, 5 and B 1, 2,
  , 9, 10
• The relation with the rule (1,1), (2,3), (4,4),
  (3,4), (5,6)
• The relation with the rule (1,1), (3,2), (4,4),
  (3,4), (5,6)

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When is a binary relation a function?

• Example 3. Let A 1, 2, 3, 4, 5 and B 3, 4,
  5, 6.

1 2 3 4 5
1 2 3 4 5
3 4 5 6
3 4 5 6
A B
A B
Relation 1
Relation 2
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Inverse of a relation

• Given a relation R from A to B, the inverse is a
  relation R-1 from B to A such that
• (y, x) ? R if and only if (x, y) ? R
• Example
• Let A 1, 2, 3, 4, 5 and B a, b, c, d. The
  relation R (1,a), (2,c), (1,d), (3,b) has
  inverse R-1 (a,1), (c,2), (d,1), (b,3).

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Inverse of a relation

• Practice. Given the diagram of the relation R on
  the left, draw the diagram of the relation R-1.

2
2
1
1
3
3
0
4
0
4
5
5
7
7
6
6
R-1
R
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Inverse of a function

• If R is a function, is R-1 necessarily a
  function?
• Draw the diagram for an example where the answer
  is yes.
• Draw the diagram for an example where the answer
  is no.
• Whats the difference?