Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if we remove the assumption "A is non-empty"? Is the state- ment in (a) still correct? Prove or provide a counterexample.
Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if we remove the assumption "A is non-empty"? Is the state- ment in (a) still correct? Prove or provide a counterexample.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.4: Prime Factors And Greatest Common Divisor
Problem 17E
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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.
Transcribed Image Text:Let A, B be set with A non-empty.
(a) Prove that if A C B, then A, B are not disjoint.
(b) What if we remove the assumption “A is non-empty”? Is the state-
ment in (a) still correct? Prove or provide a counterexample.
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