Logic and Truth Tables

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Questions and Answers

What is the symbolic representation of 'Today is Friday and it is raining'?

~q ∧ r

p ∧ q (correct)

p ∨ q

~p ∧ ~q

In a truth table, when both statements p and q are true, what is the value of p ∧ q?

Undefined

Cannot be determined

T (correct)

How would you express 'It is not raining and I am going to a movie' in symbolic form?

~q ∧ r (correct)

~q ∨ r

q ∧ r

~p ∨ ~r

What does the statement 'Today is not Friday and it is not raining' translate to in symbolic form?

~p ∧ ~q Signup and view all the answers

Which symbolic statement represents 'The game will not be played in MSU or the Sultans are favored to win'?

~p ∨ s Signup and view all the answers

What is the value of p ∨ q when both p and q are false in a truth table?

F Signup and view all the answers

If p represents 'The game will be shown on ABS-CBN', how is the statement 'The game will not be shown on GMA' expressed symbolically?

~r Signup and view all the answers

What is the truth set of the negation for the statement involving $p_x: x + 4 = 4$ and $q_x: x^2 < 5$?

{−4, −3, 3, 4} Signup and view all the answers

How do you symbolize the statement 'I am going to the basketball game or I am going to a movie'?

s ∨ r Signup and view all the answers

How many elements are there in the set $Q$ derived from the statement $q_x: x^2 < 5$?

5 Signup and view all the answers

What does the statement 'For all members in Club 2, the member has red hair' represent?

Universal statement Signup and view all the answers

What values are included in the complement $P_C$ derived from the statement $p_x: x - 1 < 2$?

{3, 4} Signup and view all the answers

Which of the following correctly represents the logical structure of the negation of the statement involving $p_x$ and $q_x$?

∼$p_x$ ∧ ∼$q_x$ Signup and view all the answers

What is the truth set of the negation given the conditions $x - 1 > 2$ and $3x - 2 ≠ 0$?

{3, 4} Signup and view all the answers

Which of the following is true about existential statements?

They assert truth for at least one element in a set. Signup and view all the answers

Which value of $x$ satisfies the equation $p_x: x + 4 = 4$?

0 Signup and view all the answers

What is the truth value of the statement '∀𝑥 ∈ {1,2,3}: 𝑥² is less than 10'?

True Signup and view all the answers

What quantifier is used in the statement '∃𝑥 ∈ {10,15,20}: 3𝑥 + 1 is odd'?

Existential quantifier Signup and view all the answers

Which statement is false regarding the values assigned to '𝑥' in the equation '3𝑥 + 1 is odd'?

The statement is true for x = 15. Signup and view all the answers

The truth value of the statement '∀𝑥 ∈ {1,2,3}: 𝑥 + 𝑦 is prime' is?

False Signup and view all the answers

What is the correct negation of the false statement 'All dogs are mean'?

Some dogs are not mean. Signup and view all the answers

If the statement '2𝑥 - 5 = 5' has a truth value of false, what can be concluded?

There are no values of x that satisfy it. Signup and view all the answers

When using the concept of ordered pairs for '∀', which values of x and y could potentially make 'x + y is prime' true?

Certain combinations of pairs need to be evaluated. Signup and view all the answers

What might be a common misconception about negating universal statements?

The negation is always the opposite statement. Signup and view all the answers

What is the converse of the statement 'If you are good in Mathematics, then you are good in logic'?

If you are good in logic, then you are good in Mathematics. Signup and view all the answers

What is the contrapositive of the statement 'If I get the job, then I will rent the apartment'?

If I do not rent the apartment, then I did not get the job. Signup and view all the answers

Which derived statement is formed by negating both parts of the conditional statement?

Inverse Signup and view all the answers

If the conditional statement is true, which of the following derived statements is always true?

Contrapositive only Signup and view all the answers

For the statement 'If x is an odd integer, then x^2 + 2 is even', what is the inverse?

If x is even, then x^2 + 2 is odd. Signup and view all the answers

Which derived conditional is denoted by ~q → ~p?

Contrapositive Signup and view all the answers

What is the truth value relationship between a conditional statement and its contrapositive?

They have the same truth value. Signup and view all the answers

What is the contrapositive of 'If we have a quiz today, then we will not have a quiz tomorrow'?

If we do have a quiz tomorrow, then we do not have a quiz today. Signup and view all the answers

What is the negation of the statement 'All movies are worth the price of admission'?

Some movies are not worth the price of admission. Signup and view all the answers

Which of the following statements is the definition of the negation of 'Some X are Y'?

No X are Y. Signup and view all the answers

Which statement correctly negates the statement 'Some students are not permitted to submit their research assignment'?

All students are permitted to submit their research assignment. Signup and view all the answers

What is the correct negation of the statement 'No odd numbers are divisible by 2'?

Some odd numbers are divisible by 2. Signup and view all the answers

Given the statement '∀x ∈ {0,1,2,3}: x^3 - 1 is an odd integer', what is the truth value of its negation?

True for at least one value of x. Signup and view all the answers

What is the negation of the statement '∃x, y ∈ {-1, 0, 1}: x = y + 1'?

For all x, y ∈ {-1, 0, 1}, x ≠ y + 1. Signup and view all the answers

If the statement is 'Some airports are open', what is the proper negation?

No airports are open. Signup and view all the answers

For the statement 'There exists a dog that is not mean', which of the following is the negation?

All dogs are mean. Signup and view all the answers

Study Notes

Truth Tables

Truth Value for 𝒑 ∧ 𝒒:
- If 𝒑 is true and 𝒒 is true, then 𝒑∧𝒒 is true
- If 𝒑 is true and 𝒒 is false, then 𝒑∧𝒒 is false
- If 𝒑 is false and 𝒒 is true, then 𝒑∧𝒒 is false
- If 𝒑 is false and 𝒒 is false, then 𝒑∧𝒒 is false
Truth Value for 𝒑 ∨ 𝒒:
- If 𝒑 is true and 𝒒 is true, then 𝒑∨𝒒 is true
- If 𝒑 is true and 𝒒 is false, then 𝒑∨𝒒 is true
- If 𝒑 is false and 𝒒 is true, then 𝒑∨𝒒 is true
- If 𝒑 is false and 𝒒 is false, then 𝒑∨𝒒 is false
Example in text: The statement “I've fallen and I can't get up" means the same as "I've fallen but I can't get up."
- If 𝑝 is "I've fallen" and 𝑞 is "I can get up", then the conjunction can be symbolized as p ∧ ~q.

Symbolic Representation of Statements

Example: Given that 𝑝: Today is Friday. 𝑞: It is raining. 𝑟: I am going to a movie. 𝑠: I am not going to the basketball game.
- "Today is Friday and it is raining" can be written symbolically as 𝑝 ∧ 𝑞.
- "It is not raining and I am going to a movie" can be written symbolically as ∼ 𝑞 ∧ 𝑟.
- "I am going to the basketball game or I am going to a movie" can be written symbolically as ∼ 𝑠 ∨ 𝑟.
- "Today is not Friday and it is not raining" can be written symbolically as ~𝑝 ∧ ~𝑞.

Negation of Statements

Example: The statement “𝑥 + 4 ≠ 4 and 𝑥 2 > 5” can be written as 𝑝𝑥 ∧∼ 𝑞𝑥.
- To find the truth set, find the values of 𝑥 that satisfy the open sentences.
- Example: For 𝑝𝑥 : 𝑥 + 4 = 4, the only value of 𝑥 that makes it true is 0.
- For 𝑞𝑥 : 𝑥 2 < 5, the only values of 𝑥 that make it true are −2, −1, 0, 1, and 2.
- The truth set of the negation is the intersection of the complements of the truth sets of 𝑝 and 𝑞.

Universal and Existential Statements

Universal Statement: States a property that is true for all members of a set.
- Example: "All positive numbers are greater than zero."
Existential Statement: States that there exists at least one member of a set that has a certain property.
- Example: "There exists a member of Club 1 such that the member has red hair."

Negation of Quantified Statements

Negation of ∀𝑥: 𝑝: ~(∀𝑥: 𝑝) ≡ ∃𝑥: ~𝑝
Negation of ∃𝑥: 𝑝: ~(∃𝑥: 𝑝) ≡ ∀𝑥: ~𝑝

Negation Examples

Example: Some airports are open.
- Negation: No airports are open.
Example: All movies are worth the price of admission.
- Negation: Some movies are not worth the price of admission.
Example: No odd numbers are divisible by 2.
- Negation: Some odd numbers are divisible by 2.
Example: Some students are not permitted to submit their research assignment.
- Negation: All students are permitted to submit their research assignment.

Conditional Statements

Conditional Statement: A statement that can be written in the form "If 𝑝, then 𝑞."
- 𝑝 is the antecedent.
- 𝑞 is the consequent.
Converse: 𝑞 → 𝑝
Inverse: ~𝑝 → ~𝑞
Contrapositive: ~𝑞 → ~𝑝

Example: If you are good in Mathematics, then you are good in logic.
- Converse: If you are good in logic, then you are good in Mathematics.
- Inverse: If you are not good in Mathematics, then you are not good in logic.
- Contrapositive: If you are not good in logic, then you are not good in Mathematics.
Example: If I get the job, then I will rent the apartment.
- Converse: If I rent the apartment, then I get the job.
- Inverse: If I do not get the job, then I will not rent the apartment.
- Contrapositive: If I do not rent the apartment, then I did not get the job.
Example: If 𝑥 is an odd integer, then 𝑥 2 + 2 is even.
- Converse: If 𝑥 2 + 2 is even, then 𝑥 is an odd integer.
- Inverse: If 𝑥 is an even integer, then 𝑥 2 + 2 is odd.
- Contrapositive: If 𝑥 2 + 2 is odd, then 𝑥 is an even integer.
Example: If we have a quiz today, then we will not have a quiz tomorrow.
- Converse: If we will not have a quiz tomorrow, then we have a quiz today.
- Inverse: If we do not have a quiz today, then we will have a quiz tomorrow.
- Contrapositive: If we will have a quiz tomorrow, then we do not have a quiz today.

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Description

This quiz covers the fundamentals of truth tables and symbolic representation of statements in logic. Explore how conjunctions and disjunctions work through practical examples and scenarios. Test your understanding of logical expressions and their truth values.