Logic+ 09 - Formal Logic
We will use the following symbols for logical operations:
∨ disjunction (logical OR),
∧ conjunction (logical AND),
~ negation (logical NOT),
→ implication (if...then).
Logical manipulation with true or false statements (conjunction, disjunction, negation) can be modeled using the analogy with integer numbers, where true statement acts like a positive integer number (usually, number 1), false statement acts like zero, conjunction acts like multiplication, disjunction acts like addition, negation of positive number is zero and negation of zero is some positive number (usually, number 1).
Here is an illustration of the analogy between logical operations and operations on integer numbers (letter T stands for a true statement, letter F stands for a false statement):
T ∧ T = T; 1 · 1 = 1
T ∧ F = F; 1 · 0 = 0
F ∧ F = F; 0 · 0 = 0
T ∨ T = T; 1 + 1 = 2
T ∨ F = T; 1 + 0 = 1
F ∨ F = F; 0 + 0 = 0
~T = F; ~1 = 0
~F = T; ~0 = 1
Another important analogy is between logical operations and operations on sets.
In particular,
Conjunction is analogous to intersection of two sets.
Disjunction is analogous to union of two sets.
Negation is analogous to complement of a set.
We can also use this analogy with sets to model the logical operation of implication.
When statement A implies statement B
Both analogies described above are very useful in analyzing the truth value of a complicated logical statement.
Problem A
Consider the following statements:
S1 - It's raining
S2 - I like to walk in a park
S3 - I sing
S4 - I forgot my umbrella
S5 - I like dogs
S6 - I like cats
S7 - I read books
S8 - I bake a pie
Express symbolically the following statement:
Q1 - When it's raining, I don't like to walk in the park.
Q2 - When it's not raining, I like to walk in the park.
Q3 - When it's not raining, I like to walk in the park and sing.
Q4 - I like to walk in the park and sing.
Q5 - It's raining and I did not forget my umbrella.
Q6 - I like both dogs and cats.
Q7 - I like dogs, but not cats.
Q8 - I don't like neither dogs nor cats.
Q9 - When it's raining, I either read a book or bake a pie.
Q10 - It's raining, so I read a book and bake a pie.
Answer A
Q1 = S1 → (~S2)
Q2 = (~S1) → S2
Q3 = (~S1) → (S2 ∧ S3)
Q4 = S2 ∧ S3
Q5 = S1 ∧ (~S4)
Q6 = S5 ∧ S6
Q7 = S5 ∧ (~S6)
Q8 = (~S5) ∧ (~S6)
Q9 = S1 → (S7 ∨ S8)
Q10 = S1 ∧ S7 ∧ S8
Problem B
Given the following statements:
S1(true) - Prague is the capital of Chech Republic
S2(true) - There are 100 cents in a dollar
S3(false) - China is a European country
S4(false) - All adult population of the Earth is literate
Evaluate (true or false) the following statements:
Q1 = S1 ∧ S2
Q2 = S1 ∧ (~S2)
Q3 = S1 ∨ S2
Q4 = S1 ∨ (~S2)
Q5 = S1 ∨ (~S1)
Q6 = S1 ∧ (~S1)
Q7 = S1 ∧ S3
Q8 = S1 ∧ (~S3)
Q9 = S1 ∨ S3
Q10 = S1 ∨ (~S3)
Q11 = S3 ∨ (~S3)
Q12 = S3 ∧ (~S3)
Q13 = S4 ∧ S3
Q14 = S4 ∧ (~S3)
Q15 = S4 ∨ S3
Q16 = S4 ∨ (~S3)
Answer B
Q1 is True
Q2 is False
Q3 is True
Q4 is True
Q5 is True
Q6 is False
Q7 is False
Q8 is True
Q9 is True
Q10 is True
Q11 is True
Q12 is False
Q13 is False
Q14 is False
Q15 is False
Q16 is True
Problem C
Consider a basic implication
"If A then B" A→B, where
A (antecedent): two numbers are integers
B (consequent): their sum is an integer
Converse: B→A
Inverse: ~A→~B
Contrapositive: ~B→~A
Q1: Basic A→B: If A is a true statement, is B a true statement?
Q2: Converse B→A: If B is a true statement, is A a true statement?
Q3: Inverse ~A→~B: If A is a false statement, is B a false statement?
Q4: Contrapositive ~B→~A: If B is a false statement, is A a false statement?
Answer C
Q1: Yes, B is True. True → True. From a true antecedent statement only true consequent statement can follow.
Q2: Cannot tell, truth or false consequent statement has no influence on antecedent one.
Q3: Cannot tell, from a false antecedent statement anything can follow, true or false.
Q4: Yes, A is False, because it cannot be true, it would contradict Q1. If A→B is True, ~B→~A is also True.
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