MTH 305 Test 1: Formal Logic, Symbolic Representation, and Proofs

This document presents the solution to MTH 305 Test 1, focusing on formal logic. The solution includes translating compound statements into symbolic form, constructing and analyzing truth tables to determine tautologies, and applying pseudocode. It also involves justifying steps in a proof sequence within propositional logic and using propositional logic to prove argument validity. Furthermore, the document addresses quantifiers, predicates, and validity, determining truth values, and writing predicate wffs. Finally, it includes negating statements, providing a comprehensive overview of the topics covered in the test and offering a valuable resource for students studying formal logic and related mathematical concepts. The assignment covers topics like symbolic representation, truth tables, propositional logic, and predicate logic.

Test 1
MTH 305
JULY 7, 2019

Name :
MTH 305
Test 1: Formal Logic
Statements, Symbolic Representation, and Tautologies
1) Let A, B, C, and D be the following statements:
A: The villain is French.
B: The hero is American.
C: The heroine is British.
D: The movie is good.
Translate the following compound statements into symbolic form, (a propositional wff).
a. The hero is American and the movie is good.
Sol.
B∧ D
b. If the movie is good, then either the hero is American or the heroine is British.
Sol.
D →(B ∨C )
c. The hero is not American but the villain is French.
Sol.
∼ B∧ A

2) Construct a truth table for the following wff.
Is this wff a tautology?
(A V B)' ↔ A’ Λ B'
[Please show 8 columns. Last column is: (A V B)' ↔ A’ Λ B')]
Sol.
A B A’ B’ A ∨ B ( A ∨ B)' ( A ∨ B ) ' ↔ A'
( A ∨ B ) ' ↔ A' ∧ B '
F F T T F T T T
F T T F T F F F
T F F T T F T T
T T F F T F T F
No, it is not a tautology.
3) Use the pseudocode on p 21 #35 to find the output values for inputs (x values): 2.0,
5.0, 6.0, 10.0
Outputs:
Sol.
Page number not available
Propositional Logic
4) Justify each step in the proof sequence.
[A → (B → C)] Λ (A V D') Λ B Λ D → C
1. A → (B → C) : ∼ A ∨(B → C) = A ∨ ( ∼ B∨ C ) =∼ A ∨∼ B ∨C
2. A V D’ : F ∨∼ B ∨C ∨ ∼ D
3. B : F ∨C ∨∼ D

4. D :F ∨C
5. D' V A : D’ V A
6. D → A : D’ V A
7. A: F
8. B → C : B’ V C
9. C : C
5) Use propositional logic to prove that the argument is valid.
A’ Λ (B → A) → B’
1. A' ∧B' ∨ A
2.( A' ∨ A ) ∧ B '
3. 1 ∧B'=B '
Quantifiers, Predicates, and Validity
6) What is the truth value of each of the following wff in the interpretation where the
domain consists of integers?
a. ("x)(x² > 0) : T
b. ($x)($y)(x² > y) : F
c. ("x)("y)[(x < y) → (y > x)] : T
7) Using the predicate symbols shown and the appropriate quantifiers, write each English
language statement as a predicate wff. (The domain is the whole world.)
B(x) is “x is a ball.”
R(x) is “x is round.”
S(x) is “x is a soccer ball.”
a. All balls are round. ( ∀ x ) B → R

b. Some soccer balls are round. (∃ x ) S ∧ B → R
c. Some soccer balls are not round.(∃ x ) S ∧ B → R '
d. Every soccer ball is round.(∀ x)S ∧ B → R
8) Write the negation of the following statement:
“Every student eats pizza.” : No student eats pizza.