The document outlines an assignment related to formal logic, covering topics such as statements, truth tables, and well-formed formulas. It includes tasks for writing symbolic forms of statements, constructing truth tables, and analyzing truth values under given conditions. Additionally, it addresses identifying well-formed formulas and tautologies, as well as substitution instances of logical expressions.

1. DMS Sunita M Dol
Walchand Institute of Technology, Solapur Page 1
Assignment No. 3
Topics covered:
• Introduction
• Statements and Notation
• Connectives
– Negation,
– Conjunction,
– Disjunction,
– Conditional,
– Bi-conditional
• Statement formulas and truth tables
• Well-formed formulas
• Tautologies
1. Using statements
R: Mark is rich
H: Mark is happy
Write the following statements in symbolic form
a. Mark is poor but happy.
b. Mark is rich or unhappy.
c. Mark is neither rich nor happy.
d. Mark is poor or he is both rich and unhappy.
2. Construct the truth table for the following formulas
a. ¬ (¬P ∨ ¬Q)
b. ¬ (¬P ∧ ¬Q)
c. P ∧ (P ∨ Q)
d. P ∧ (Q ∧ P)
e. (¬ P ∧ (¬ Q ∧ R)) ∨ (Q ∧ R) ∨ (P ∧ R)
f. (P ∧ Q) ∨ (¬P ∧ Q) ∨ (P ∧ ¬Q) ∨ (¬P ∧ ¬Q)
g. (Q ∧ (P → Q)) → P
h. ¬ (P ∨ (Q ∧ R)) ↔ ((P ∨ Q) ∧ (P ∨ R))
3. Given the truth values of P and Q as T and those of R and S as F, find the
truth values of the following

2. DMS Sunita M Dol
Walchand Institute of Technology, Solapur Page 2
a. P ∨ (P ∧ Q)
b. (P ∧ (Q ∧R )) ∨ ¬ ((P ∨ Q) ∧ (R ∨ S))
c. (¬ (P ∧ Q) ∨ ¬R) ∨ (((¬P ∧ Q) ∨ ¬R) ∧ S)
d. ((¬P ∧ Q) ∨ ¬R) ∨ ((Q ↔ ¬P) → (R ∨ ¬S))
e. (P ↔ R) ∧ (¬Q → S)
f. (P ∨ (Q → (R ∧ ¬P))) ↔ (Q ∨ ¬S)
4. For what truth values will the following statement be true?
‘It is not the case that houses are cold or haunted and it is false that cottages
are warm or houses are ugly’
5. From the formulas given below select those which are well-formed and
indicate which ones are tautologies and contradictions
a. (P → (P ∨ Q))
b. ((P → (¬P)) → ¬P)
c. ((¬Q ∧ P) ∧ Q)
d. ((P → (Q → R)) → ((P → Q) → (P → R)))
e. ((¬P → Q) → (Q → P)))
f. ((P ∧ Q) ↔ P)
6. Produce the substitution instances of the following formulas for the given
substitutions
a. (((P → Q) → P) → P) ; substitute (P → Q) for P and ((P ∧ Q) → R)
for Q
b. ((P → Q) → (Q → P)) ; substitute Q for P and (P ∧ ¬P) for Q
7. Determine the formulas which are substitution instance of other formulas in
the list and give the substitutions
a. (P → (Q → P))
b. ((((P → Q) ∧ (R → S)) ∧ (P ∨ R)) → (Q ∨ S))
c. (Q → ((P → P) → Q))
d. (P → ((P → (Q → P)) → P))
e. ((((R → S) ∧ (Q → P)) ∧ (R ∨ Q)) → (S ∨ P))