Reference no: EM132682997

Question 1:

Let P , Q and R propositions defined as follows:
P : I am a UCLA student.
Q: My SE Principle module is completed. R: It is end of the semester.

Write each of the following propositions as logical expression involving P , Q, and R:

1. I am a UCLA student and my SE Principle module is completed.

2. It is end of the semester and my SE Principle module is not completed.

3. If my SE Principle module is not completed, then it is not end of the semester.

4. If it is end of the semester, then my SE Principle module is completed.5. If it is end of semester and my SE Principle module is not completed, then I am not a UCLA student.

Question 2:

Translate the following into symbolic form:
1. Every UCLA student is smart .

2. Someone at UCLA is smart.

3. Nobody can ignore Peter

4. Everyone likes someone.

5. There is a student who is loved by every other student..

Question 3:

1. Let's consider a propositional language where: A ="Angelo comes to the party",
B ="Bruno comes to the party", C ="Carlo comes to the party", D ="David comes to the party".
Formalize the following sentences:

• Angelo comes to the party while Bruno doesn't.

• Either Carlo comes to the party, or Bruno and David don't come.

• If Angelo and Bruno come to the party, then Carlo comes provided that David doesn't come.

• Carlo comes to the party if Bruno and Angelo don't come, or if David comes.

Question 4:

The following statement describes a simple flat tyre problem:

The spare should be at the trunk and the flat is at the axle. If the spare is at the trunk, remove the spare from the trunk. If the flat is at axle, remove the flat from the axle. Put on spare to the axle. Remove the flat from the ground and get on the trunk.

Considering possible actions such as At, Remove, PutOn, ... represent the solution of flat tyre problem in at least 10 correct arguments.

Question 5:

The following statement describes a simple elevator/lift system:

The elevator is moving between floors or is stationary at a floor. If the elevator is stationary then the brakes are applied. The brakes are not applied therefore the elevator is moving between floors. If the elevator is stationary or the breaks are applied, then elevator are not moving and vice versa.

i. Represent the above statement using propositions.

ii. Test the validity of the argument by constructing a truth table.

Question 6:

In a recent small survey a group of workers entering a building were asked which floor they had cause to visit to do their work. A set of people (P ) identified by their first names, P = Jack, John, Bob, Matt, Dan, were asked to choose from the three floors in the buildings F = {GROUND, ONE, T W O}. There is also a set of ages for the workers given by A = {x ∈ N | 16 < x < 70}. In ages: Jack is 20, John is 25, Bob is 40, Matt is 55 and Dan is 65.

i. Write down the set of members of the Cartesian product P × F

ii. Write down the relation, which is a subset of P × A, which relates people to age.

Question 7:

Let U = {x ∈ N : x 6 15}. Let A = {x : x is even}, B = {x : x < 8}, and C = {x : x is divisible by 3}. Depict the sets on a Venn diagram.

Hence, write down the following sets in enumerated form:

i. A ∩ B ii. B ∪ C
iii. A ∪ B iv. (A ∪ B)

∩ C v. (A ∩ C) ∪ A ∪ B ∪ C

Question 8:

Let U = {x ∈ N : x ≤ 12}. Let A = {x : x is odd}, B = {x : x > 7}, and C = {x : x is divisible by 3}. i. With the help of Venn diagrams, show that for finite sets A, B and C

| A ∪ B ∪ C | = | A | + | B | + | C | - | A ∩ B | - | B ∩ C | - | A ∩ C | + | A ∩ B ∩ C |

Question 9:

For the purpose of error detection, numeric codes (such as Passport numbers) often include a final ‘check digit'.

Suppose a numeric code consists of a string of 9 digits x1x2 . . . x9, followed by a final check digit x10 defined to be the rightmost decimal digit of x1 + 2x2 + 3x3 + . . . + 9x9.

i. Verify that 5241562639 is a valid code.

ii. Validate 2516238674, whether is a valid code or not.

iii. Let X be the set of all strings of 9 digits, let Y be the set of all digits, and let f : X → Y be the function that assigns the correct check digit to each string, for example f(251623867) = 4. State, giving reasons, whether f is one-to-one and whether f is onto.

iv. If an error is made in keying in a code, will the check digit always detect it? Explain, with reference to your answer to (ii).