Part I: Suppose you are developing a statistical database in which information about professional football teams and records are stored.
Consider the following 2 sets of data that list football teams and quarterbacks:
D = {Jets, Giants, Cowboys, 49’ers, Patriots, Rams, Chiefs}
Q = {Tom Brady, Joe Namath, Troy Aikman, Joe Montana, Eli Manning}
1.Using D as the domain and Q as the range, show the relation between the 2 sets, with the correspondences based on which players are (or were) a member of which team(s). (Show the relation in the following forms:
◦Set of ordered pairs
◦Directional graph
2.Is the relation a function? Explain.
3.Now, use set Q as the domain, and set D as the range. Show the relation in the following forms:
◦Set of ordered pairs
◦Directional graph
4.Is the relation a function? Explain.
For the first problem the ordered pairs are (Jets, Namath) (Giants, Manning) (Cowboys, Aikman) (49ers, Montana) (Chiefs, Montana) (Patriots, Brady). The directional map is scanned and attached. This is a function. No one element of the domain matches more than one element in the range.
For the second problem, reverse the ordered pairs and find that it is no longer a function, since one element of the domain (Joe Montana) matches with two elements of the range (49ers and Chiefs).
1. Set of ordered pairs:
The relation between the sets D and Q can be represented as a set of ordered pairs, where each pair consists of a football team from set D and the corresponding quarterback from set Q.
{(Jets, Joe Namath), (Giants, Eli Manning), (Cowboys, Troy Aikman), (49’ers, Joe Montana), (Patriots, Tom Brady)}
2. Directional graph:
The relation can also be represented as a directional graph, where each football team from set D is connected to the corresponding quarterback from set Q. The graph would have 5 edges with the football teams as the starting point and quarterbacks as the endpoint.
3. Set of ordered pairs:
If we use set Q as the domain and set D as the range, the relation can be represented as a set of ordered pairs, where each pair consists of a quarterback from set Q and the football team(s) they are (or were) a member of from set D.
{(Tom Brady, Patriots), (Joe Namath, Jets), (Troy Aikman, Cowboys), (Joe Montana, 49’ers), (Eli Manning, Giants)}
4. Is the relation a function? Yes, the relation is a function.
For a relation to be a function, each element from the domain (set Q) must correspond to only one element from the range (set D). In the given relation, each quarterback from set Q is associated with only one football team from set D. Therefore, the relation is a function.