Search & Filtering
Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.
a) The product of two negative real numbers is positive.
b) The difference of a real number and itself is zero.
c) Every positive real number has exactly two square roots.
d) A negative real number does not have a square root that is a real number. e) Every non-zero real number has a unique reciprocal.
Implement the following questions using Python. Make use of conditional and print statements. Make sure to clearly define your variables and don’t forget to indent your code.
Note: Solutions without indentation will be awarded no marks.
The logician Raymond Smulyan describes an island containing two types of people:
knights who always tell the truth and knaves who always lie. You visit the island and are approached by two natives who speak to you as follows:
• A says: B is a knight.
• B says: A and I are of opposite type.
What are A and B?
Hint: You are required to solve 4 puzzles from this problem statement. Think!
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.
a) ∃x ∀y(x + y = y)
b) ∀x ∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))
c) ∃x ∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))
d) ∀x ∀y((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))
Implement the following questions using Python. Make use of conditional and print statements.
Make sure to clearly define your variables and don’t forget to indent your code.
Note: Solutions without indentation will be awarded no marks.
Part a:
If compound X is boiling, then its temperature must be at least 150◦C.” Assuming that this
statement is true, which of the following must also be true?
a. If the temperature of compound X is at least 150◦C, then compound X is boiling.
b. If the temperature of compound X is less than 150◦C, then compound X is not boiling.
c. Compound X will boil only if its temperature is at least 150◦C.
d. If compound X is not boiling, then its temperature is less than 150◦C.
e. A necessary condition for compound X to boil is that its temperature be at least 150◦C.
The notation: ∃! x P(x)
means “There exists a unique x such that P(x)”.
If the domain consists of all integers, what are the truth values of these statement?
1. ∃! x(x > 1)
2. ∃! x(x
2 = 1)
3. ∃! x(x + 3 = 2x)
4. [∃! xP(x)] → [∃xP(x)]
5. [∀xP(x)] → [∃! xP(x)]
6. [∃! x~P(x)] → [~∀xP(x)]
7. ∃! x(x = x + 1)
8. ~(∃! xP(x)) → ∀xP(x)
9. (∃xP(x) ∧ ∃xQ(x)) → ∃x (P(x) ∧ Q(x))
10. (∀xP(x) ∨ ∀xQ(x)) → ∀x (P(x) ∧ Q(x))
Determine if the following argument is valid or if it exhibits the converse or the inverse error. Use symbols to write the logical form of argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made.
If at least one of these two numbers is divisible by 6,
then the product of these two numbers is divisible by 6.
Neither of these two numbers is divisible by 6.
∴ The product of these two numbers is not divisible by 6.
1)Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.
a) ∀x(C(x) → F(x))
b) ∀x(C(x) ∧ F(x))
c) ∃x(C(x) → F(x))
2) Somie, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Somie on the basis of the following statements:
a. Socko: Lefty killed Somie.
b. Fats: Muscles didn’t kill Somie.
c. Lefty: Muscles was shooting craps with Socko when Somie was knocked off.
d. Muscles: Lefty didn’t kill Somie.
Who did kill Somie?
Show that the following pair of statements are logically equivalent with proper explanation:
1. ∀x (P(x) ∧ Q(x)) and ∀x P(x) ∧ ∀x Q(x)
2. ¬ (∀x P(x)) and ∃x ¬P(x)
3. ¬ (∃x P(x)) and ∀x ¬P(x)
show that p ↔ q and (p^q) v (¬p^¬q) are logical equivalent
I would like to ask for help with discrete mathematics, especially two's complement
- Using two’s complement representation throughout your calculation, find 31 – 19.
- Using Euclidean algorithm, find the GCD of 123 and 411.
