Q2 b with full explanation
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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Question
Q2 b
with full explanation
![1. Let f : X →Y be a function.
(a) Prove that f(B1 N B2) = f(B1) n f(B2) for any B1, B2, E P(Y).
(b) Prove that f(A N A2) s f(A1)n f (A2) for any A1, A2 E P(X). Give an example
of a specific function f and sets A1, A, such that f(A1) n f(A2) Z f(AN Az2)
2. Let f : X →Y be a function.
(a) Prove that f is injective + f is injective + f is surjective.
(b) Prove that f is surjective + f is surjective + f is injective.
(Hint: when proving statements of the form P Q R, it is often easier to prove
P = Q = R → P than to prove P Q and Q R)
3. Prove Corollary 10.2.2 from class: for any n E Z4, if X1,. .. , Xn are finite sets which
are pairwise disjoint, then
Ux =EX:|.
i=1
i=1
n
(Recall that U X; = X1 U X, U ...UX,
{x]x € X; for some i e {1,2,..,n}} by
i=1
definition).
4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then
|AN B| > 3.
5. Prove the following theorem from class:
Theorem 1. Suppose that X and Y are nonempty finite sets with |X| < [Y| and
f : X → Y is a function. Then f is not a surjection.
6. Prove the following theorem from class:
Theorem 2. Suppose that X and Y are finite sets with |X| = |Y\. Then a function
f: X → Y is injective if and only if it is surjective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2Fdad87217-5b11-41ba-9285-c8e8a04d3b79%2F2iqi2o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let f : X →Y be a function.
(a) Prove that f(B1 N B2) = f(B1) n f(B2) for any B1, B2, E P(Y).
(b) Prove that f(A N A2) s f(A1)n f (A2) for any A1, A2 E P(X). Give an example
of a specific function f and sets A1, A, such that f(A1) n f(A2) Z f(AN Az2)
2. Let f : X →Y be a function.
(a) Prove that f is injective + f is injective + f is surjective.
(b) Prove that f is surjective + f is surjective + f is injective.
(Hint: when proving statements of the form P Q R, it is often easier to prove
P = Q = R → P than to prove P Q and Q R)
3. Prove Corollary 10.2.2 from class: for any n E Z4, if X1,. .. , Xn are finite sets which
are pairwise disjoint, then
Ux =EX:|.
i=1
i=1
n
(Recall that U X; = X1 U X, U ...UX,
{x]x € X; for some i e {1,2,..,n}} by
i=1
definition).
4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then
|AN B| > 3.
5. Prove the following theorem from class:
Theorem 1. Suppose that X and Y are nonempty finite sets with |X| < [Y| and
f : X → Y is a function. Then f is not a surjection.
6. Prove the following theorem from class:
Theorem 2. Suppose that X and Y are finite sets with |X| = |Y\. Then a function
f: X → Y is injective if and only if it is surjective.
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