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Chapter 1: Problem 4

You have been put in charge of drawing up the schedule for a basketball league. This league involves eight teams, each of which must play each of the other seven teams exactly two times: once at home and once on the road. Think of a reasonable language for this situation. What constants would you need? Do you need any relation symbols? Function symbols? It would be nice if your finished schedule did not have any team playing two games on the same day. Can you think of a way to state this using the formal symbols that you have chosen? Can you express the sentence which states that each team plays every other team exactly two times?

Short Answer

Expert verified
Use team constants, play relation symbols, and constraints to ensure each team plays others twice without playing two games a day.

Step by step solution

01

Identify Constants

In this scheduling problem, we can identify the following constants: the set of 8 teams labeled as \( T_1, T_2, \ldots, T_8 \), the set of days for the league \( D_1, D_2, \ldots \), where the full schedule is completed. Each game can also be seen as a constant, potentially labeled by the match and occasion, like \( G_{a,b,h} \) where a match occurs between team \( T_a \) and \( T_b \) at the venue of \( T_a \).
02

Consider Relation Symbols

We need relation symbols to describe the structure of the schedule. The relation \( ext{Plays}(G, D) \) might be used to indicate that game \( G \) is scheduled on day \( D \). Another relation symbol, \( ext{Opponents}(T_i, T_j) \), could be used to indicate that team \( T_i \) plays against team \( T_j \).
03

Explore Function Symbols

Function symbols are not necessarily required because the relationships and rules can be captured using our defined constants and relations. However, if we chose to include them, a function \( ext{HomeTeam}(G) \) could return the home team for game \( G \).
04

Formulate Constraints

Ensure that each team plays every other team exactly twice. Using our symbols, for each team \( T_i \), there should exist exactly two games \( G_1 \) and \( G_2 \) such that \( ext{Opponents}(T_i, T_j) \) is true and occurs exactly once at home and once away. Mathematically, this can be expressed as: \( \forall T_i,T_j, i eq j, \exists G_1, G_2: \text{Plays}(G_1, D_1) \land \text{Plays}(G_2, D_2) \land \text{HomeTeam}(G_1) = T_i \land \text{HomeTeam}(G_2) = T_j \).
05

Avoid Scheduling Conflicts

For ensuring that no team plays two games on the same day, one can state that for any team \( T_i \), and any given day \( D_x \), there can be at most one game such that \( ext{Plays}(G, D_x) \land (T_i \in (\text{HomeTeam}(G) \lor \text{Opponents}(G, T_i))) \) is true.
06

Translate to Scheduling

Use the relations and the language structure to draft an actual schedule by assigning each game a day, respecting the constraints we've defined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constants
In the context of a scheduling problem for a basketball league, constants serve as fixed values or entities inherently involved, which do not change. Think of constants as the foundation upon which our schedule is built.
For instance, we can define the following constants:
  • Teams: Label each of the 8 teams uniquely, such as \( T_1, T_2, \ldots, T_8 \) to easily reference them throughout the schedule.
  • Days: To manage when games occur, we create a set of constants for days the league will take place, such as \( D_1, D_2, \ldots \).
  • Games: Each specific match can be a constant, which might be expressed as \( G_{a,b,h} \). This notation tells us that team \( T_a \) plays team \( T_b \) at team \( T_a\)'s venue.
These constants provide clear points of reference throughout the scheduling process, enabling clarity in communication and decision-making.
Relation Symbols
In mathematical logic, relation symbols are used to describe relationships or interactions between constants and elements. They are crucial for expressing the dynamics of a scheduling problem.
Several relation symbols could be beneficial here:
  • \( \text{Plays}(G, D) \): This relation indicates that a game \( G \) is scheduled on day \( D \). It helps map games to specific dates within the league's calendar.
  • \( \text{Opponents}(T_i, T_j) \): This states that team \( T_i \) is matched against team \( T_j \). It provides a straightforward way to capture the essential match-up between teams for structuring the league.
These relation symbols provide a logical structure to the scheduling formulation, making it easier to set rules, like ensuring each team plays every other team twice.
Scheduling Problem
The concept of a scheduling problem involves arranging events, activities, or tasks within a specified timeframe to meet certain criteria or constraints. In our context of a basketball league, it refers to organizing games in a way that respects various conditions.
The criteria or constraints for the basketball league can include:
  • Each Team Plays Every Other Team: Formulate using the constraints that each team must play every other team twice, once home and once away. This involves creating games such that \( \text{Opponents}(T_i, T_j) \) is fulfilled with two different games having team \( T_i \) home once and away once.
  • No Overlapping Games: Ensure no team plays two games on the same day. This is crucial for logistical purposes and is expressed as at most one game involving any team per day \( D \).
Addressing these aspects of the scheduling problem allows for a coherent league schedule, ensuring fair play and adequate rest for all teams.

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Most popular questions from this chapter

Show that if \(t\) is variable-free, then \(t\) is always substitutable for \(x\) in \(\phi\).

Carefully write out the symbols that you would want to have in a language \(\mathcal{L}\) that you intend to use to write statements of elementary algebra. Indicate which of the symbols are constant symbols, and the arity of the function and relation symbols that you choose. Now write out another language, \(\mathcal{M}\) (i.e., another list of symbols) with the same number of constant symbols, function symbols, and relation symbols that you would not want to use for elementary algebra. Think about the value of good notation.

Suppose that \(\mathcal{L}\) is \(\\{0, f, g\\}\), where 0 is a constant symbol, \(f\) is a binary function symbol, and \(g\) is a 4-ary function symbol. Use induction on complexity to show that every \(\mathcal{L}\) -term has an odd number of symbols.

Show that \(x\) is always substitutable for \(x\) in \(\phi\).

One more bit of shorthand. Assume that the language \(\mathcal{L}\) contains the binary relation symbol \(\in\), which you are intending to use to mean the elementhood relation (so \(p \in q\) will mean that \(p\) is an element of \(q\) ). Often, it is the case that you want to claim that \(\phi(x)\) is true for every element of a set \(b .\) Of course, to do this you could write $$(\forall x)[(x \in b) \rightarrow \phi(x)]$$ We will abbreviate this formula as $$(\forall x \in b)(\phi(x))$$ Similarly, \((\exists x \in b)(\phi(x))\) will be an abbreviation for the formula \((\exists x)[(x \in b) \wedge \phi(x)] .\) Notice that this formula has a conjunction where the previous formula had an implication!. We do that just to see if you are paying attention. (Well, if you think about what the abbreviations are supposed to mean, you'll see that the change is necessary. We'll have to do something else just to see if you're paying attention.) Now suppose that \(\mathfrak{A}\) is a structure for the language of set theory. So \(\mathcal{L}\) has only this one binary relation symbol, \(\in\), which is interpreted as the elementhood relation. Suppose, in addition, that $$A=\\{u, v, w,\\{u\\},\\{u, v\\},\\{u, v, w\\}\\}$$ In particular, notice that there is no element \(x\) of \(A\) such that \(x \in x .\) Consider the sentence $$(\forall y \in y)(\exists x \in x)(x=y)$$ Is this sentence false in \(\mathfrak{A}\) ?
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