Homework-1, Adaptation course in DM

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Homework-1, Adaptation course in DM - 2022

Dear students,

Here are the tasks of the first homework for you to make sure that you have not missed anything in the lectures and also for your additional practice. Problems with stars give you bonus points. After you send your form there will be a comment for your answers and the correct answers. Just compare your answers with mine. This form may mark with red even the correct answers. Do not worry, I will check it myself later.

Once you send the form the obtained point is your grade for this homework.

Email *

My name and surname *

1. Which of the following sentences is/are statement(s)? *

1 point

How to solve quadratic equation ?

Sum of two squares.

Set is the mathematical model for a collection of different things.

3x + 2 = 7

2. Which of the following statements is/are atomic statement(s)? (That means they cannot be split into parts)
*

1 point

If number 25 is a perfect square then 3 + 7 = 12.

Logical connective “and” is called conjunction and connective “or” is called disjunction.

Neither Kate nor Pete are going for a walk

Number 0 is not natural

3(a) Let P and Q be statements P - "Jack passes math", Q - "Jill passes math". Translate the following statement into symbols “Both Jack and Jill have passes math”. *

1 point

¬(P ∧ Q)

P ∧ Q

P ∨ Q

¬P ∧ Q

3(b) Translate the following statement into symbolic language "If Jack passes math, then Jill can't pass it."
*

1 point

¬P --> Q

P --> ¬Q

¬P ∧ Q

P --> Q

3(c) Translate into words “P ∨ Q”. *

1 point

Jack will pass or fail math.

Either Jill or Jack will pass the math.

Both Jack and Jill will pass math.

Jill will pass or fail math.

3(d) Translate into words P<->Q. *

1 point

If Jack passes the math, then Jill will pass the math.

If Jill passes the math, then Jack will pass the math.

Jack will pass math if and only if Jill passes.

3(d)* Translate into words “¬(P ∧ Q) → Q”. *

1 point

3(e)* Translate into words "(P ∧ ¬ P) ∧ Q" *

1 point

4(а). Denote predicate: A(x) = “3x + 1 is an even number”. Find A(3). *

1 point

True

False

Other

Other:

4(b). Denote predicate: A(x) = “3x + 1 is an even number”. Find A(8). *

1 point

True

False

Other

5(a) Denote the predicate B(x). What can you say about the statement ∃xB(x) if you know that the statement B(5) is true?
*

1 point

True

False

Can be both true and false

5(b) Denote the predicate B(x). What can you say about the statement ∀x B(x) if you know that the statement B(5) is true?
*

1 point

True

False

Can be both true and false

6. Use the truth table to check that the statements ¬(P → Q) and P ∧ ¬Q are logically equivalent.
*

1 point

Yes, they are equivalent.

No, they are not equivalent.

7. Check the truth of the statement for the set of natural numbers N = {1,2,3,4, ....} ∃x(x=45)∨(x=6)
*

1 point

True

False

8. Check the truth of the statement "∀x x^2 >0" (the symbol ^ denotes exponentiation) for the set of natural numbers N = {1,2,3,4, ....}
*

1 point

True

False

9. Construct the negation of the statement "∀x x^2 >0". Will it be true or false?
*

1 point

"∀x x^2 > 0", Истинно

"∀x x^2 < 0", Ложно

"∃x x^2 < 0", Ложно

"∃x x^2 ⩽ 0", Ложно

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