Chapter 2 · Computational Thinking & Algorithms

2.9Quantifiers in Predicate Logic

Two symbols, for all and there exists, that tell us how many objects make a statement true.

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In topic 2.8 you met predicates, like "x is greater than 5". A predicate alone does not say how many objects we are talking about. Is it every x? Or just one? Quantifiers answer that question. They are small words, "for all" and "there exists", with their own symbols.

The book's examples are "All students are hardworking" and "Some books are interesting." The word "All" needs one quantifier. The word "Some" needs the other. This is why quantifiers make predicate logic more powerful than propositional logic.

Universal Quantifier (for all)

The universal quantifier talks about every object in a set. Its symbol is an upside-down A.

The book's example is "∀x, x > 0". Read it as: for all x, x is greater than 0. A "for all" statement is strict. It is true only if no object in the set breaks it. This makes it useful for rules and in mathematics and logic proofs.

Existential Quantifier (there exists)

The existential quantifier needs only one object that fits. Its symbol is a backwards E.

The book's example is "∃x, x is even": there exists at least one x that is even. We use this quantifier when some object meets the condition, but not necessarily all of them. It helps us talk about possibilities and specific examples, so it is important for problem solving and reasoning.

Figure 2.8 in the book, "Universal and Existential Quantifier", puts the two side by side. The box "∀x" leads to "For all x, P(x)". The box "∃x" leads to "For some x, P(x)".

Figure 2.8 redrawn: ∀x leads to For all x, P(x), with the example ∀x, x > 0; ∃x leads to For some x, P(x), with the example ∃x, x is even

	Universal (for all)	Existential (there exists)
Symbol	∀	∃
True for	all objects in a given set	at least one object in a set
Book example	∀x, x > 0	∃x, x is even

Using Quantifiers in Logical Statements

A quantifier is placed in front of a predicate. "∀x, P(x)" says P(x) is true for every x. "∃x, P(x)" says P(x) is true for at least one x.

We can also use two quantifiers together. The book's example is "∀x ∃y, x < y". Read it from left to right: for every x, there exists a y, such that x is less than y. In other words, whatever x you pick, a bigger y can be found. Each x may get a different y.

With quantifiers we can state rules, conditions and relationships clearly. In the next topic, 2.10, we use them to write the rules of a library.

Practice this topic

All MCQs of this chapter with answers

Short questions

1.What are quantifiers in predicate logic?2 marks
Answer
From the book: "Quantifiers are used in predicate logic to describe how many objects satisfy a condition. They help us make general statements or statements about specific objects." The chapter Summary says: "A quantifier is used in predicate logic to show how many objects a statement applies to, such as for all or there exists."
2.What is the universal quantifier? Give its symbol and one example.3 marks
Answer
From the book: "The universal quantifier is used to indicate that a statement is true for all objects in a given set (as shown in Figure 2.8). It is denoted by the symbol "∀". For example, "∀x, x > 0" means "For all x, x is greater than 0.""
3.What is the existential quantifier? Give its symbol and one example.3 marks
Answer
From the book: "The existential quantifier is used to indicate that a statement is true for at least one object in a set (as shown in Figure 2.8). It is denoted by the symbol "∃". For example, "∃x, x is even" means "There exists at least one x that is even.""
4.Differentiate between "∀x, P(x)" and "∃x, P(x)".2 marks
Answer
From the book: ""∀x, P(x)" means the predicate P(x) is true for all x, while "∃x, P(x)" means P(x) is true for at least one x." Figure 2.8 reads them as "For all x, P(x)" and "For some x, P(x)".
5.What does the statement "∀x ∃y, x < y" mean?2 marks
Answer
From the book: "Quantifiers can also be used together in complex statements. For instance, "∀x ∃y, x < y" means for every x there exists a y that is greater than x."
6.Why do quantifiers make predicate logic more powerful than propositional logic?3 marks
Answer
From the book: "Quantifiers make predicate logic more powerful than propositional logic. Using quantifiers, we can write statements like "All students are hardworking" or "Some books are interesting." Quantifiers help represent knowledge clearly and precisely."
7.When is the existential quantifier used, and why is it important?3 marks
Answer
From the book: "This quantifier is used when a condition is satisfied by some object but not necessarily all. It helps in expressing possibilities and specific examples. The existential quantifier is important for problem solving and reasoning."

Long questions

1.Explain the universal quantifier and the existential quantifier. Give the symbol, meaning and an example of each, and describe Figure 2.8.8 marks
Answer
Cover both sub-headings and the figure, using the book's text:
1. Universal Quantifier (for all) (verbatim): "The universal quantifier is used to indicate that a statement is true for all objects in a given set (as shown in Figure 2.8). It is denoted by the symbol "∀"." Example: ""∀x, x > 0" means "For all x, x is greater than 0."" Uses: "The universal quantifier is used to make general statements about every object in a domain. It is important in mathematics and logic proofs. Using "for all" ensures that no object in the set violates the statement."
2. Existential Quantifier (there exists) (verbatim): "The existential quantifier is used to indicate that a statement is true for at least one object in a set (as shown in Figure 2.8). It is denoted by the symbol "∃"." Example: ""∃x, x is even" means "There exists at least one x that is even."" Uses: "This quantifier is used when a condition is satisfied by some object but not necessarily all. It helps in expressing possibilities and specific examples."
3. Figure 2.8: Universal and Existential Quantifier: "∀x" leads to "For all x, P(x)"; "∃x" leads to "For some x, P(x)".
4. Close with the TIDBIT: "Quantifiers like "for all" or "there exists" help us express general rules or possibilities clearly."
Aim for 200-300 words.
2.What are quantifiers? Explain how quantifiers are combined with predicates and used together in logical statements.6 marks
Answer
Use the 2.9 opening paragraph and the sub-heading "Using Quantifiers in Logical Statements":
1. Definition (verbatim): "Quantifiers are used in predicate logic to describe how many objects satisfy a condition. They help us make general statements or statements about specific objects."
2. Why they matter: "Quantifiers make predicate logic more powerful than propositional logic." Examples of statements: "All students are hardworking" or "Some books are interesting."
3. Combined with predicates (verbatim): ""∀x, P(x)" means the predicate P(x) is true for all x, while "∃x, P(x)" means P(x) is true for at least one x."
4. Used together: ""∀x ∃y, x < y" means for every x there exists a y that is greater than x."
5. Conclusion: "Using quantifiers allows us to express rules, conditions, and relationships clearly." "Quantifiers help represent knowledge clearly and precisely."
Aim for 150-250 words.

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