Test 1: Formal Logic

This document is Test 1 for the course MTH 305. It covers topics such as statements, symbolic representation, and tautologies in formal logic. The document includes exercises on translating compound statements into symbolic form, constructing truth tables, using pseudocode, justifying proof sequences, and using propositional logic to prove arguments. It also covers quantifiers, predicates, and validity, including determining truth values of wffs and writing English language statements as predicate wffs. The document concludes with an exercise on writing the negation of a statement.

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.