30. Show that 2 − is irrational.
31. Show that is irrational.
32. Prove or disprove the following statements.
(a) The sum of a rational and an irrational number must be irrational.
(b) The sum of two positive irrational numbers must be irrational.
(c) The product of a nonzero rational and an irrational number must be
irrational.
33. Show that every positive integer can be expressed as the product of prime
numbers.
34. Prove that the set of all prime numbers is infinite.
35. A prime pair consists of two primes that differ by two. There are many
prime pairs, for example, 11 and 13, 17 and 19, etc. Prime triplets are three
numbers n ≥ 2, n + 2, n + 4 that are all prime. Show that the only prime
triplet is (3, 5, 7).
1.2 Three Basic Concepts
Three fundamental ideas are the major themes of this book: languages,
grammars, and automata. In the course of our study we will explore many
results about these concepts and about their relationship to each other. First, we
must understand the meaning of the terms.
Languages
We are all familiar with the notion of natural languages, such as English and
French. Still, most of us would probably find it difficult to say exactly what the
word “language” means. Dictionaries define the term informally as a system
suitable for the expression of certain ideas, facts, or concepts, including a set of
symbols and rules for their manipulation. While this gives us an intuitive idea of
what a language is, it is not sufficient as a definition for the study of formal
languages. We need a precise definition for the term.
31. Show that is irrational.
32. Prove or disprove the following statements.
(a) The sum of a rational and an irrational number must be irrational.
(b) The sum of two positive irrational numbers must be irrational.
(c) The product of a nonzero rational and an irrational number must be
irrational.
33. Show that every positive integer can be expressed as the product of prime
numbers.
34. Prove that the set of all prime numbers is infinite.
35. A prime pair consists of two primes that differ by two. There are many
prime pairs, for example, 11 and 13, 17 and 19, etc. Prime triplets are three
numbers n ≥ 2, n + 2, n + 4 that are all prime. Show that the only prime
triplet is (3, 5, 7).
1.2 Three Basic Concepts
Three fundamental ideas are the major themes of this book: languages,
grammars, and automata. In the course of our study we will explore many
results about these concepts and about their relationship to each other. First, we
must understand the meaning of the terms.
Languages
We are all familiar with the notion of natural languages, such as English and
French. Still, most of us would probably find it difficult to say exactly what the
word “language” means. Dictionaries define the term informally as a system
suitable for the expression of certain ideas, facts, or concepts, including a set of
symbols and rules for their manipulation. While this gives us an intuitive idea of
what a language is, it is not sufficient as a definition for the study of formal
languages. We need a precise definition for the term.