Chapter 2 · Computational Thinking & Algorithms

2.10Applying Predicate Logic to Real-World Problems

Using predicates, quantifiers and reasoning to describe a library system and make decisions in it.

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So far, predicate logic has been about symbols. This topic puts it to work. The book uses one example all the way through: a library system, with its students, books and teachers. We describe who is connected to whom, write the library's rules, and then reason with them.

Predicate logic lets us describe objects, their properties and the links between them. Once the knowledge is written clearly, we can study a problem, make decisions and find answers in a logical way.

Representing Relationships Using Predicates

A predicate with two variables can describe a link between two objects. The book gives two:

Borrows(x, y) means "student x borrows book y,"
Teaches(t, s) means "teacher t teaches student s."

In the book's words, these predicates "clearly describe how students, books, and teachers are connected." The book calls the people and things "entities". Writing their links as predicates makes the whole system easier to organise and understand.

Writing Logical Statements With Quantifiers

Relationships describe single facts. Rules need quantifiers (topic 2.9). The book gives two library statements:

Statement	Meaning (book wording)
∀x, Student(x) → Borrows(x, Book)	"every student borrows a book."
∃y, Book(y) ∧ Available(y)	"there exists at least one book that is available."

The arrow → is read "if ... then". The symbol ∧ is read "and". So the first line says: for every x, if x is a student, then x borrows a book. The second says: there is at least one y that is a book and is available.

The first statement is a general rule about everyone. The second is about a specific case. Quantifiers let us describe both.

The book's CLASS ACTIVITY asks you to write rules like "All students must attend class" using universal quantifiers. Try it in the same style as the first formula.

Reasoning and drawing conclusions

Now we can reason. Take a rule and a fact:

Rule: "∀x, Student(x) → HasLibraryCard(x)" (every student has a library card).
Fact: "Ali is a student."
Conclusion: "Ali has a library card."

The rule is true for every student, and Ali is a student, so it must be true for Ali.

The library system as predicates: Teaches(t, s) and Borrows(x, y) link teacher, student and book; two quantified rules; a rule plus the fact Ali is a student give the conclusion Ali has a library card

This kind of reasoning helps the library make decisions, such as whether a student can borrow a book. Because every conclusion follows the same rules, the answers are correct and consistent. That makes the system reliable and easy to manage.

Practice this topic

All MCQs of this chapter with answers

Short questions

1.How can predicate logic be used to solve real-world problems?3 marks
Answer
From the book: "Predicate logic can be used to solve real-world problems by representing knowledge clearly and systematically. It helps us model situations involving objects, their properties, and relationships. Using predicate logic, we can analyze problems, make decisions, and find solutions logically."
2.How can predicates represent relationships? Give the book's two examples.3 marks
Answer
From the book: "Predicates can represent relationships between objects in the real world. For example, Borrows(x, y) means "student x borrows book y," and Teaches(t, s) means "teacher t teaches student s." These predicates clearly describe how students, books, and teachers are connected."
3.Write the meaning of "∀x, Student(x) → Borrows(x, Book)" and "∃y, Book(y) ∧ Available(y)".2 marks
Answer
From the book: "∀x, Student(x) → Borrows(x, Book) means "every student borrows a book." ∃y, Book(y) ∧ Available(y) means "there exists at least one book that is available."" Note: in the first formula "Book" is used like the name of one object; a fully formal version is ∀x (Student(x) → ∃y (Book(y) ∧ Borrows(x, y))). In the exam, write the book's formula and meaning as printed.
4.How do we draw conclusions using predicate logic? Give the book's example.3 marks
Answer
From the book: "Using predicate logic, we can draw conclusions from given facts. For example, if "∀x, Student(x) → HasLibraryCard(x)" and "Ali is a student," then we conclude "Ali has a library card.""
5.Why is logical reasoning useful in a library system?2 marks
Answer
From the book: "This logical reasoning helps in decision making, such as checking whether a student can borrow a book. It ensures correct and consistent conclusions, making the system reliable and easy to manage."
6.How do quantifiers help in writing rules about the library?2 marks
Answer
From the book: "Quantifiers help us write general rules about the library." "These statements help define rules and conditions in a precise way. Quantifiers allow us to describe both general situations and specific cases in the library system."

Long questions

1.Explain predicate logic and quantifiers. Discuss how predicate logic is applied to real-world problems.8 marks
Answer
This book question spans three topics. Parts 1 and 2 come from topics 2.8 and 2.9 (marked); part 3 is this topic, 2.10.
1. Predicate logic (topic 2.8, verbatim): "Predicate logic is an extension of propositional logic. It allows us to express more detailed statements about objects and their properties." "A predicate is a statement that can be true or false depending on the values of the variables." Example from 2.8: Tall(Ali).
2. Quantifiers (topic 2.9, verbatim): "Quantifiers are used in predicate logic to describe how many objects satisfy a condition." Universal quantifier "∀" (for all): ""∀x, x > 0" means "For all x, x is greater than 0."" Existential quantifier "∃" (there exists): ""∃x, x is even" means "There exists at least one x that is even."" Figure 2.8: "For all x, P(x)" and "For some x, P(x)".
3. Applying predicate logic to real-world problems (topic 2.10, verbatim): "Predicate logic can be used to solve real-world problems by representing knowledge clearly and systematically. It helps us model situations involving objects, their properties, and relationships."
  
  Relationships: Borrows(x, y) means "student x borrows book y," and Teaches(t, s) means "teacher t teaches student s." "Using such relationships, we can model the working of the library system."
  
  Rules with quantifiers: "∀x, Student(x) → Borrows(x, Book) means "every student borrows a book." ∃y, Book(y) ∧ Available(y) means "there exists at least one book that is available.""
  
  Reasoning: if "∀x, Student(x) → HasLibraryCard(x)" and "Ali is a student," then we conclude "Ali has a library card." "This logical reasoning helps in decision making, such as checking whether a student can borrow a book."
  
  DO YOU KNOW?: "Predicate logic can model family trees, social networks, and even computer programs!"
Aim for 300-400 words.
2.Explain, with the book's library example, how predicates, quantifiers and reasoning are used to model a real-world system.6 marks
Answer
Cover the three sub-headings of 2.10 in order, using the book's text:
1. Representing Relationships Using Predicates: "Predicates can represent relationships between objects in the real world." Borrows(x, y) means "student x borrows book y," and Teaches(t, s) means "teacher t teaches student s." "These predicates clearly describe how students, books, and teachers are connected." "This makes it easier to organize and understand interactions between different entities."
2. Writing Logical Statements With Quantifiers: "Quantifiers help us write general rules about the library." The two formulas and their meanings ("every student borrows a book"; "there exists at least one book that is available"). "Quantifiers allow us to describe both general situations and specific cases in the library system."
3. Reasoning and drawing conclusions: "Using predicate logic, we can draw conclusions from given facts." The Ali example (student, so has a library card). "It ensures correct and consistent conclusions, making the system reliable and easy to manage."
Aim for 200-300 words.

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