COT 3541 – Midterm Questions And
Full Set Correct Solutions |Update
2026
Atomic propositions - Ans--An atomic proposition is a proposition whose
truth or falsity does not rely on the truth or falsity of any other proposition.
Conjunction - Ans--A sentence with an ∧ connective such as P ∧ Q
Disjunction - Ans--A sentence with an ∨ connective
Precedence of connectives - Ans--First precedence Parenthesis
Second precedence Negation
Third precedence Conjunction (And)
Fourth precedence Disjunction (Or)
Fifth precedence Implication
Sixth precedence Biconditional
Commutativity - Ans--• P ∧ Q = Q ∧ P
•P∨Q=Q∨P
Associativity - Ans--• (P ∧ Q) ∧ R = P ∧ (Q ∧ R)
• (P ∨ Q) ∨ R = P ∨ (Q ∨ R)
Identity element - Ans--• P ∧ True = P
• P ∨ True = True
Distributive - Ans--• P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)
• P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R)
DE Morgan's Law - Ans--• ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
• ¬ (P ∨ Q) = (¬P) ∧ (¬Q)
, Inverse - Ans--p → q will be ¬p → ¬q
Is the negation of both the hypothesis and the conclusion
"If you have enough money, then you will buy a new house." will be "If you do
not have enough money, then you will not buy a new house."
Converse - Ans--Is created by interchanging the hypothesis and the
conclusion.
If q, then p
If you have enough money, then you will buy a newhouse." will be "If you buy
a new house, then you have enough money.
Contra-positive - Ans--Is created by interchanging the hypothesis and the
conclusion of the inverse statement.
p → q will be ¬q → ¬p
If you have enough money, then you will buy a newhouse." will be "If you do
not buy a new house, then you did not have enough money."
Duality principle - Ans--For any TRUE statement
The dual statement achieved by interchanging unions into intersections
and vice versa.
• And, interchanging Universal set into NULL set and vice versa• Will be
TRUE
• Hint: If dual of any statement is the statement itself, it is said self-dual
statement.• Example:• The Dual of (P ∩ Q) ∪ R will be (P ∪ Q) ∩ R
Normal forms - Ans--It is possible to convert any proposition into two normal
• Conjunctive normal form
• Disjunctive normal form
Conjunctive normal form - Ans--Is a clausal normal form of a logical formula
consisting of a conjunction of disjunctions; it can also be described as an
AND of ORs or a product of sums
If we put disjunctive clauses together with ∧ or ∩
(P ∨ Q) ∧ (P ∨ R) ∧ (Q ∨ R ∨ S)
(P ∨ R) ∧ (¬Q ∨ ¬R) ∧ Q
Full Set Correct Solutions |Update
2026
Atomic propositions - Ans--An atomic proposition is a proposition whose
truth or falsity does not rely on the truth or falsity of any other proposition.
Conjunction - Ans--A sentence with an ∧ connective such as P ∧ Q
Disjunction - Ans--A sentence with an ∨ connective
Precedence of connectives - Ans--First precedence Parenthesis
Second precedence Negation
Third precedence Conjunction (And)
Fourth precedence Disjunction (Or)
Fifth precedence Implication
Sixth precedence Biconditional
Commutativity - Ans--• P ∧ Q = Q ∧ P
•P∨Q=Q∨P
Associativity - Ans--• (P ∧ Q) ∧ R = P ∧ (Q ∧ R)
• (P ∨ Q) ∨ R = P ∨ (Q ∨ R)
Identity element - Ans--• P ∧ True = P
• P ∨ True = True
Distributive - Ans--• P ∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R)
• P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R)
DE Morgan's Law - Ans--• ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
• ¬ (P ∨ Q) = (¬P) ∧ (¬Q)
, Inverse - Ans--p → q will be ¬p → ¬q
Is the negation of both the hypothesis and the conclusion
"If you have enough money, then you will buy a new house." will be "If you do
not have enough money, then you will not buy a new house."
Converse - Ans--Is created by interchanging the hypothesis and the
conclusion.
If q, then p
If you have enough money, then you will buy a newhouse." will be "If you buy
a new house, then you have enough money.
Contra-positive - Ans--Is created by interchanging the hypothesis and the
conclusion of the inverse statement.
p → q will be ¬q → ¬p
If you have enough money, then you will buy a newhouse." will be "If you do
not buy a new house, then you did not have enough money."
Duality principle - Ans--For any TRUE statement
The dual statement achieved by interchanging unions into intersections
and vice versa.
• And, interchanging Universal set into NULL set and vice versa• Will be
TRUE
• Hint: If dual of any statement is the statement itself, it is said self-dual
statement.• Example:• The Dual of (P ∩ Q) ∪ R will be (P ∪ Q) ∩ R
Normal forms - Ans--It is possible to convert any proposition into two normal
• Conjunctive normal form
• Disjunctive normal form
Conjunctive normal form - Ans--Is a clausal normal form of a logical formula
consisting of a conjunction of disjunctions; it can also be described as an
AND of ORs or a product of sums
If we put disjunctive clauses together with ∧ or ∩
(P ∨ Q) ∧ (P ∨ R) ∧ (Q ∨ R ∨ S)
(P ∨ R) ∧ (¬Q ∨ ¬R) ∧ Q
