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Let f A → B and g : B → C be functions. For each of the following, prove or provide a counterexample. (a) If 9 of is injective, is f always injective? (b) If go f is injective, is g always injective? (c) If 9 of is surjective, is ƒ always surjective? (d) If go f is surjective, is g always surjective?

Let f A → B and g : B → C be functions. For each of the following, prove or provide a counterexample. (a) If 9 of is injective, is f always injective? (b) If go f is injective, is g always injective? (c) If 9 of is surjective, is ƒ always surjective? (d) If go f is surjective, is g always surjective?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let f A → B and g : B → C be functions. For each of the following, prove or provide a counterexample. (a) If go f is injective, is f always injective? (b) If gof is injective, is g always injective? (c) If go f is surjective, is f always surjective? (d) If g of is surjective, is always surjective?
Transcribed Image Text:Let f A → B and g : B → C be functions. For each of the following, prove or provide a counterexample. (a) If go f is injective, is f always injective? (b) If gof is injective, is g always injective? (c) If go f is surjective, is f always surjective? (d) If g of is surjective, is always surjective?
For this homework, you can use the following facts. If it is a definition, you should mention "by definition" or "by definition of XXX (e.g. even)". If it is not a definition, you do NOT have to cite. 1. Basic algebra, such as 2 + 2 = 4, 1-3 = -2, 2-4 = 8. This includes subtracting an integer, or dividing by a non-zero rational/real/complex numbers. 2. Common knowledge of whether numbers are integer/rational, e.g. is not an integer, is not rational, all integers are rational, all rational numbers are real, all real numbers are complex, etc. 3. Associative law of addition: (a+b)+c= a + (b+c), and associative law of multiplication: (ab)c = a(be), for all a, b, c € C. 4. Commutative law of addition: a+b=b+a, and commutative law of multiplication: abba, for all a, b, c € C. 5. Distributive law: For all a, b, c € C, we have (a + b)(c+d) = ac+ad+ be + bd. In particular, (a + b)² = a² + 2ab + b². 6. An integer n is even if there exists an integer k such that n = 2k. 7. An integer n is odd if there exists an integer k such that n = 2k + 1. 8. All integers are either even or odd. 9. A real number az is positive if a > 0, and negative if x < 0. 10. All real numbers are either positive, negative, or 0. 11. Let A, B be subsets of a universal set U. (a) We say A is a subset of B, denoted by ACB, if for all z € A, we have z € B. (b) An element is in AUB, the union of A and B, if z € A or € B. (c) An element z is in An B, the intersection of A and B, if x € A and € B. (d) An element z EU is in Aº, the complement of A in U, if æ & A. 12. We denote by the empty set. 13. Let A, B be sets. Then (a) An BCA; (b) ACAUB. 1 14. A "function" f: A → B is a well-defined function, or simply a function, if for all z EA, there is a unique y E B such that f(x) = y. (There are quotes here as f is not actually a function if it is not well-defined. ) 15. A function f: A → B is injective if for all ₁,22 € A, if f(x₁) = f(x₂), then #1 = 22- 16. A function f : A → B is surjective if for all y € B, there exists € A such that f(x) = y. 17. A function f : A → B is bijective if it is injective and surjective.
Transcribed Image Text:For this homework, you can use the following facts. If it is a definition, you should mention "by definition" or "by definition of XXX (e.g. even)". If it is not a definition, you do NOT have to cite. 1. Basic algebra, such as 2 + 2 = 4, 1-3 = -2, 2-4 = 8. This includes subtracting an integer, or dividing by a non-zero rational/real/complex numbers. 2. Common knowledge of whether numbers are integer/rational, e.g. is not an integer, is not rational, all integers are rational, all rational numbers are real, all real numbers are complex, etc. 3. Associative law of addition: (a+b)+c= a + (b+c), and associative law of multiplication: (ab)c = a(be), for all a, b, c € C. 4. Commutative law of addition: a+b=b+a, and commutative law of multiplication: abba, for all a, b, c € C. 5. Distributive law: For all a, b, c € C, we have (a + b)(c+d) = ac+ad+ be + bd. In particular, (a + b)² = a² + 2ab + b². 6. An integer n is even if there exists an integer k such that n = 2k. 7. An integer n is odd if there exists an integer k such that n = 2k + 1. 8. All integers are either even or odd. 9. A real number az is positive if a > 0, and negative if x < 0. 10. All real numbers are either positive, negative, or 0. 11. Let A, B be subsets of a universal set U. (a) We say A is a subset of B, denoted by ACB, if for all z € A, we have z € B. (b) An element is in AUB, the union of A and B, if z € A or € B. (c) An element z is in An B, the intersection of A and B, if x € A and € B. (d) An element z EU is in Aº, the complement of A in U, if æ & A. 12. We denote by the empty set. 13. Let A, B be sets. Then (a) An BCA; (b) ACAUB. 1 14. A "function" f: A → B is a well-defined function, or simply a function, if for all z EA, there is a unique y E B such that f(x) = y. (There are quotes here as f is not actually a function if it is not well-defined. ) 15. A function f: A → B is injective if for all ₁,22 € A, if f(x₁) = f(x₂), then #1 = 22- 16. A function f : A → B is surjective if for all y € B, there exists € A such that f(x) = y. 17. A function f : A → B is bijective if it is injective and surjective.
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