2.1.1 - Introduction to Mathematic Logic, Propositions or Statements.

• A) A question that can be true or false
• B) A command that has a specific meaning
• C) A declarative sentence that is either true or false, but not both simultaneously
• D) An exclamation that expresses a feeling

• A) The numerical value of a mathematical expression
• B) The meaning of a statement in everyday language
• C) The truth or falsity of a statement
• D) The complexity of a logical expression

• A) "One plus two equals three."
• B) "Close the door."
• C) "One plus two equals five."
• D) "The sky is blue."

• A) True, False, and Unknown
• B) Yes and No
• C) True (T) and False (F)
• D) Positive and Negative

• A) Symbols like &, |, ~
• B) Numbers like 1, 2, 3
• C) Letters such as p, q, r
• D) Greek letters like α, β, γ

• A) A statement can be true and false at the same time.
• B) A proposition is a type of question.
• C) A proposition must have a clear truth value.
• D) Exclamations are valid propositions in logic.

• A) Multi-valued logic
• B) Three-valued logic
• C) Binary logic
• D) Two-valued logic

• A) A complete sentence like "The Earth is round."
• B) A symbol like +
• C) A letter such as p or q
• D) A numerical value like 5

• A) It helps in solving complex calculus problems.
• B) It provides the theoretical basis for areas like artificial intelligence and digital logic design.
• C) It replaces the need for algebra.
• D) It simplifies the study of biology.

• A) Questions are always propositions.
• B) Propositions are always questions.
• C) Questions can sometimes be propositions.
• D) Questions are not propositions.

• A) Practical methods for performing arithmetic
• B) Theoretical basis for areas such as artificial intelligence, digital logic, and design
• C) A way to solve differential equations
• D) Techniques for graphic design

• A) "What is your name?"
• B) "Stop talking!"
• C) "It is raining outside."
• D) "Wow, that's amazing!"

• A) Imperative
• B) Exclamatory
• C) Interrogative
• D) Declarative

• A) F or 0
• B) T or 1
• C) + or -
• D) Y or N

• A) F or 0
• B) T or 1
• C) + or -
• D) Y or N

2.1.2 - Compound Proposition and Connectives.

• A) A single propositional variable or constant
• B) A proposition made of two or more statements
• C) A statement that is always true
• D) A type of compound proposition

• A) A proposition that cannot be divided
• B) A proposition obtained from combinations of other propositions using logical operators
• C) A statement that is false
• D) A type of negation

• A) Mathematical functions
• B) Operators used to form compound propositions
• C) Types of atomic propositions
• D) Statements that are always true

• A) Negation
• B) Conjunction
• C) Disjunction
• D) Multiplication

• A) ∧
• B) ∨
• C) →
• D) ~

• A) ∧
• B) ∨
• C) →
• D) ~

• A) ∧
• B) ∨
• C) →
• D) ↔

• A) ∧
• B) ∨
• C) →
• D) ↔

• A) ∧
• B) ∨
• C) →
• D) ↔

• A) The sky is blue
• B) The sky is not blue
• C) The sky is always blue
• D) None of the above

• A) Makes a true proposition false
• B) Combines two propositions
• C) Always results in a true statement
• D) Makes a false proposition true

• A) It is true only when both p and q are false
• B) It is true only when either p or q is true
• C) It is true only when both p and q are true
• D) It is true when at least one of p or q is true

• A) It is true only when both p and q are true
• B) It is true only when both p and q are false
• C) It is true when at least one of p or q is true
• D) It is false when either p or q is true

• A) When p is false and q is true
• B) When p is true and q is false
• C) When both p and q are true
• D) When both p and q are false

• A) p is a necessary condition for q
• B) q is a sufficient condition for p
• C) p is sufficient for q
• D) p is necessary and sufficient for q

• A) When p is true and q is false
• B) When both p and q are true, or both are false
• C) When p is false and q is true
• D) When both p and q are true only

• A) p and q are both true
• B) p is true only if q is true
• C) p if and only if q
• D) p or q, but not both

• A) It is raining or it is sunny
• B) It is raining and it is cold
• C) If it is raining, then I will take an umbrella
• D) If and only if it is raining, then it is cloudy

• A) If it rains, I will stay home
• B) It is raining and it is windy
• C) It is raining or it is snowing
• D) It is raining if and only if it is cloudy

• A) It is not raining
• B) It is raining and it is cold
• C) If it rains, then I will use an umbrella
• D) It is raining or it is cold

2.1.3 - Truth Table.

• A) The logical operators used in a compound statement
• B) The truth values of a compound proposition for all possible cases
• C) The syntax of logical statements
• D) The frequency of true values in a compound statement

• A) Two
• B) Three
• C) Four
• D) Five

• A) True
• B) False
• C) Undefined
• D) Depends on the value of p

• A) True
• B) False
• C) True if q is a negation
• D) Undefined

• A) Disjunction (OR)
• B) Conjunction (AND)
• C) Negation (NOT)
• D) Implication (IF)

• A) True
• B) False
• C) It cannot be determined
• D) Depends on the operator used

• A) Table 2.1.3
• B) Table 2.1.4
• C) Table 2.1.5
• D) Table 2.1.6

• A) It makes it true
• B) It keeps it unchanged
• C) It makes it false
• D) It inverts the truth value

• A) Two
• B) Three
• C) Four
• D) Five

• A) Table 2.1.4
• B) Table 2.1.5
• C) Table 2.1.6
• D) Table 2.1.7

2.1.4

• A) q → p
• B) ~q → ~p
• C) ~p → ~q
• D) p → ~q

• A) q → p
• B) ~q → ~p
• C) ~p → ~q
• D) p → ~q

• A) q → p
• B) ~q → ~p
• C) ~p → ~q
• D) p → ~q

• A) p → q
• B) q → p
• C) ~p → ~q
• D) ~q → ~p

• A) If it rains, then the crops will grow.
• B) If it does not rain, then the crops will not grow.
• C) If the crops do not grow, then it did not rain.
• D) If the crops grow, then it rained.

• A) If the crops grow, then it rained.
• B) If the crops do not grow, then it did not rain.
• C) If it does not rain, then the crops will not grow.
• D) If it rains, then the crops will grow.

• A) Converse of p → q
• B) Contrapositive of p → q
• C) Inverse of p → q
• D) Original proposition p → q

• A) q → p
• B) ~q → ~p
• C) ~p → ~q
• D) Both B and C

• A) p → q is true
• B) p → q is false
• C) p → q is not affected
• D) None of the above

• A) "If the crops grow, then it rained."
• B) "If the crops do not grow, then it did not rain."
• C) "If it does not rain, then the crops will not grow."
• D) "It rains, and the crops do not grow."

2.1.5

• A) ( ¬ p ∧ ¬ q )
• B) ( ¬ p ∨ ¬ q )
• C) ( p ∨ q )
• D) ( p ∧ q )

• A) ( ¬ p ∧ ¬ q )
• B) ( ¬ p ∨ ¬ q )
• C) ( ¬ (p ∨ q) )
• D) ( p ∧ q )

• A) ( ¬ p )
• B) ( p )
• C) ( ¬ (¬ p) )
• D) ( ¬ (¬ (¬ p)) )

• A) ( p → ¬ q )
• B) ( ¬ p → q )
• C) ( p ∧ ¬ q )
• D) ( ¬ (p → q) )

• A) ( p ↔ ¬ q )
• B) ( ¬ p ↔ q )
• C) ( ¬ (p ↔ q) )
• D) ( ¬ p ↔ ¬ q )

• A) ( ¬ p ∧ ¬ q )
• B) ( ¬ p ∨ ¬ q )
• C) ( ¬ p → ¬ q )
• D) ( p ∨ q )

• A) ( ¬ p ∧ ¬ q )
• B) ( ¬ p ∨ ¬ q )
• C) ( ¬ p → ¬ q )
• D) ( p ∧ q )

• A) ( p ↔ ¬ q )
• B) ( ¬ p ↔ q )
• C) ( ¬ (p ↔ q) )
• D) ( p → q )

• A) ( ¬ (¬ p) ) is always false.
• B) ( ¬ (¬ p) ) is equivalent to ( p ).
• C) ( ¬ p ) is equivalent to ( ¬ (¬ p) ).
• D) ( ¬ (¬ p) ) is always true.

• A) The table where ( p → q ) is true when ( p ) is false or both ( p ) and ( q ) are true.
• B) The table where ( p → q ) is true when ( p ) is true and ( q ) is true, and false otherwise.
• C) The table where ( ¬ (p → q) ) is true only when ( p ) is true and ( q ) is false.
• D) The table where ( ¬ (p → q) ) is true for all values of ( p ) and ( q ).

2.1.6 - Algebra of Proposition.

• A) 𝑝 ∨ 𝑞 ≡ 𝑞
• B) 𝑝 ∧ 𝑞 ≡ 𝑞
• C) 𝑝 ∨ 𝑞 ≡ 𝑝
• D) 𝑝 ∧ 𝑞 ≡ 𝑝

• A) (𝑝 ∧ 𝑞) ∧ 𝑟 ≡ 𝑝 ∧ (𝑞 ∧ 𝑟)
• B) 𝑝 ∧ (𝑞 ∧ 𝑟) ≡ (𝑝 ∧ 𝑞) ∧ 𝑟
• C) (𝑝 ∨ 𝑞) ∧ 𝑟 ≡ 𝑝 ∧ (𝑞 ∧ 𝑟)
• D) (𝑝 ∧ 𝑞) ∨ 𝑟 ≡ 𝑝 ∧ (𝑞 ∨ 𝑟)

• A) 𝑝 ∨ (𝑞 ∧ 𝑟) ≡ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)
• B) 𝑝 ∧ 𝑞 ≡ 𝑞 ∧ 𝑝
• C) 𝑝 ∨ 𝑞 ≡ 𝑝 ∧ 𝑞
• D) 𝑝 ∧ (𝑞 ∨ 𝑟) ≡ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)

• A) 𝑝 ∧ (𝑞 ∨ 𝑟) ≡ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)
• B) 𝑝 ∨ (𝑞 ∧ 𝑟) ≡ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)
• C) 𝑝 ∨ (𝑞 ∧ 𝑟) ≡ (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟)
• D) 𝑝 ∧ (𝑞 ∧ 𝑟) ≡ (𝑝 ∧ 𝑞) ∧ (𝑝 ∧ 𝑟)

• A) 𝑝 ∧ 𝑇 ≡ 𝑇
• B) 𝑝 ∨ 𝐹 ≡ 𝑝
• C) 𝑝 ∧ 𝐹 ≡ 𝑝
• D) 𝑝 ∨ 𝑇 ≡ 𝑝

• A) 𝑝 ∨ ¬𝑝 ≡ 𝐹
• B) 𝑝 ∧ ¬𝑝 ≡ 𝑇
• C) ¬𝑇 ≡ 𝑇
• D) 𝑝 ∨ ¬𝑝 ≡ 𝑇

• A) ¬𝑝
• B) 𝑝
• C) ¬(¬¬𝑝)
• D) ¬(¬¬𝑝)

• A) ¬(𝑝 ∨ 𝑞) ≡ ¬𝑝 ∧ ¬𝑞
• B) ¬(𝑝 ∧ 𝑞) ≡ ¬𝑝 ∧ ¬𝑞
• C) ¬(𝑝 ∨ 𝑞) ≡ ¬𝑝 ∨ ¬𝑞
• D) ¬(𝑝 ∧ 𝑞) ≡ ¬𝑝 ∨ ¬𝑞

2.1.7 - Logical Equivalence.

• A) They have the same truth value in every possible case.
• B) They always have opposite truth values.
• C) They are tautologies.
• D) They are contradictions.

• A) ∧
• B) ∨
• C) ≡
• D) ¬

• A) Construct the truth table for Q.
• B) Construct the truth table for P.
• C) Compare the truth values of P and Q.
• D) Check if P is a tautology.

• A) Check if both propositions are tautologies.
• B) Ensure that P and Q have the same propositional variables.
• C) Compare each row to see if the truth values of P and Q match.
• D) Verify that P and Q are contradictions.

• A) P and Q are tautologies.
• B) P and Q are contradictions.
• C) P and Q are logically equivalent.
• D) P and Q are mutually exclusive.

• A) To ensure that the propositions are in standard form.
• B) To make sure that the comparisons are valid and accurate.
• C) To avoid having too many variables in the truth tables.
• D) To guarantee that the propositions are tautologies.

• A) It allows the replacement of a proposition with a simpler but equivalent one.
• B) It shows that propositions cannot be simplified.
• C) It helps to identify contradictions within the propositions.
• D) It ensures that all propositions are tautologies.

• A) Constructing a truth table for P.
• B) Constructing a truth table for Q.
• C) Checking if P and Q are tautologies.
• D) Comparing the truth values of P and Q.

• A) P and Q are logically equivalent.
• B) P and Q are contradictions.
• C) P and Q are not logically equivalent.
• D) P and Q are both tautologies.

• A) To make the proposition more complex.
• B) To simplify the proposition for easier manipulation or understanding.
• C) To increase the number of variables in the proposition.
• D) To convert the proposition into a contradiction.

2.1.8 - Tautology, Contradiction and Contingency.

• A) A statement pattern that is false for all possible combinations of truth values
• B) A statement pattern that is neither true nor false
• C) A statement pattern whose truth value is true for all possible combinations of the truth values of its prime components
• D) A statement pattern that is true for some combinations of truth values and false for others

• A) P ∧ ~P
• B) P ∨ ~P
• C) P ∧ Q
• D) P ∨ Q

• A) A statement pattern that is true for all possible combinations of truth values
• B) A statement pattern that is neither true nor false
• C) A statement pattern whose truth value is false for all possible combinations of the truth values of its prime components
• D) A statement pattern that is true for some combinations of truth values and false for others

• A) P ∨ ~P
• B) P ∧ ~P
• C) P ∧ Q
• D) P ∨ Q

• A) A statement pattern that is true for all possible combinations of truth values
• B) A statement pattern that is false for all possible combinations of truth values
• C) A statement pattern that is neither a tautology nor a contradiction
• D) A statement pattern that is true for some combinations of truth values and false for others

• A) P ∨ ~P
• B) P ∧ ~P
• C) p ∧ q
• D) ~P ∨ ~P

• A) Conjunction ( ∧ )
• B) Disjunction ( ∨ )
• C) Negation ( ~ )
• D) Implication ( → )

• A) Conjunction ( ∧ )
• B) Disjunction ( ∨ )
• C) Negation ( ~ )
• D) Implication ( → )

• A) Tautology
• B) Contradiction
• C) Contingency
• D) Equivalence

• A) P ∨ ~P
• B) P ∧ ~P
• C) (P ∨ Q) ∧ (P ∨ ~Q)
• D) P ∨ (P ∧ Q)