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Section 4.1

Section 4.1. Functions and relations. Examples of functions from algebra or pre-calculus. f : R  R with rule f ( x ) = 3 x + 1 g : Z  N with rule g ( x ) = x 2

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Section 4.1

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  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 function’s “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…

  4. Other ways to describe a function’s “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. { } {x} {y} {z} {x,y} {x,z} {y,z} {x,y,z} { } {x} {y} {z} {x,y} {x,z} {y,z} {x,y,z} 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)

  6. { } {x} {y} {z} {x,y} {x,z} {y,z} {x,y,z} 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

  7. 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 < b} captures the “less than” relationship among natural numbers. That is, R = {(0,1), (0,2), (1,2), (0,3), (1,3), (2,3), …}

  8. 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.”

  9. 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 AA

  10. 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. What’s missing in the solution below? { 2} {1 } {3 } { } {1,2,3 } {2,3 } {1,2 } { 1,3} A

  11. 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

  12. 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?

  13. 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)}

  14. 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

  15. 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)}.

  16. 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

  17. 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.” What’s the difference?

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