Logical Equivalencies and Direct Proof

Exam 3, 301 su 2020 Dr. Kimberly Vincent

Relations, Functions, Sets, and Infinity

 Write your solutions with only one question per page side of a sheet of paper. You may use both sides. (I will not read any work on this question sheet).

 Follow all directions.

 Justify all your work with good pictures, analytical work, explanations or a combination.

 USE COMPLETE SENTENCES when explaining and writing your proofs so you complete your thoughts.

When you are done with your exam sign the following

statement and include in the pdf you upload to blackboard

with your responses.

Sign and include the following statement in the pdf of your

exam:

I acknowledge that all the work submitted for

exam 1 in Math 301, summer 2020 is my own

original work.

I acknowledge that I did not share my work or

copy anyone else’s work as part of this exam.

I further acknowledge that if I shared my

answers or copied other’s work we both get

zeros on the exam.

Printed Name__________________________

Signature_____________________________

Date_______________________________

Exam 3, 301 su 2020 Dr. Kimberly Vincent

Relations, Functions, Sets, and Infinity

Turn to blank page

1. For nonempty sets A, B, C, D in some universal set U, let 𝑓:𝐴 → 𝐵, 𝑔:𝐵 → 𝐶 and ℎ:𝐶 → 𝐷 be arbitrary functions a. (( 5 pts.) What is the domain of ℎ°(𝑔°𝑓)? b. (5 pts.) What is the codomain of ℎ°(𝑔°𝑓)? a. (5 pts.) Suppose ℎ°(𝑔°𝑓) = 𝐼𝑥, where 𝐼𝑥:𝑋 → 𝑋. Which of the fours sets, A, B, C, D

must be equal to X?

Turn to blank page

2. a. Pick one of DeMorgan’s laws (you choose which) for nonempty sets A and B in some universe U. Use the same law for parts a and b.

b. (5 pts.) Draw a Venn diagram to illustrate DeMorgan’s Law c. (15 pts) Prove DeMorgan’s law for nonempty sets A and B in some universe, U.

Turn to blank page

3. (15 pts.) Let ~ be a relation on R. Suppose for 𝑥,𝑦∈ℝ, 𝑥~𝑦 if and only if 𝑥𝑦≤0. Determine (with justification; if each property below does or does not hold, prove or provide a

counterexample)

a. Is ~ reflexive, b. Is ~ symmetric c. Is ~ transitive?

Turn to blank page

4. Consider the sequence below as a function. 1/3, 1/6, 1/9, 1/12,…

a. (5pts.) What is the domain? b. (5 pts.)What is the rule of the function?

Turn to blank page

5. Let 𝑓:𝑅 − {4} → 𝑅 − {3} defined by 𝑓(𝑥) = 3𝑥

𝑥−4 .

a. (10 pts.) Is f an injection? Prove or provide a counter example. b. ( 10 pts) Is f a surjection? Prove or provide a counter example. c. (10 pts) Find the inverse relation of f. Verify that it is the inverse, as we have done in

class.

d. (10 pts) Is the inverse of f a function? Explain why it is or is not a function. Turn to blank page

6. (10 pts) Draw arrow diagram to show a function f that is a surjection but not an injection.

Now draw an arrow diagram for 𝑓−1 . Explain why 𝑓−1 won’t be a function. Turn to blank page Bonus. You will not loose points if you get the bonus wrong.

Bonus Let a function where 𝑓:𝑍5 → 𝑍5 defined by 𝑓(𝑥) = 𝑥 3(𝑚𝑜𝑑5).

a. Is f an injection? Prove or provide a counter example. b. Is f a surjection? Prove or provide a counter example. c. Find the inverse relation of f. Verify that it is the inverse, as we have done in class. d. Is the inverse of f a function? Explain why it is or is not a function.

Turn in ONE pdf with the file name “your name exam 3.pdf”