In expressions that include ⇒ and other logical operators such as ∧, ∨, and ~, the order of operations is that ⇒ is performed last while ~ is performed first.
B
Practice makes perfect
This question is testing our knowledge of connectives and logical operators. Let's look at the common symbols and their meanings.
Name
Symbol
Meaning
Order of Operations
Negation
~
NOT
1
Conjunction
∧
AND
2
Disjunction
∨
OR
3
Conditional
⇒
If/Then
4
In expressions that include ⇒ and other logical operators such as ∧, ∨, and ~, the order of operations is that ⇒ is performed last while ~ is performed first.
p ⇒ q
Let's look at the statement in option A. We will assign each letter it's truth value, using T for true and F for false.
p ⇒ q
⇕
F ⇒ T
This represents a conditional statement and is interpreted as If F, then T. This statement is only false when the hypothesis, F, is true and conclusion, T, is false. Therefore, this given statement is true.
~ q ∨ (p ∧ q)
Let's look at the statement in option B.
~ q ∨ ( p ∧ q)
⇕
~ T ∨ ( F ∧ T)
Because of the order of operations, we will interpret ~ T first. This is interpreted as not true, which indicates it is false.
~ T ∨ ( F ∧ T)
⇕
F ∨ ( F ∧ T)
Let's interpret what's in the parentheses, ( F ∧ T). The conjunction of the statements is interpreted as F and T. Thus, the statement is true only when F and T are both true. F is false, and therefore ( F ∧ T) is false.
F ∨ ( F ∧ T)
⇕
F ∨ F
The disjunction of the statements is interpreted as F or F. The statement is true only when at least one of the components is true. Therefore, F ∨ F is false. Thus, option B is the correct answer to the question, but we will still check the other options.
p ∨ q
Let's look at the statement in option C.
p ∨ q
⇕
F ∨ T
The disjunction of the statements is interpreted as F or T. The statement is true only when at least one of the components is true. T is true, and therefore this statement is true.
(p ∨ q) ∧ ~ p
Let's look at the statement in option D.
( p ∨ q) ∧ ~ p
⇕
( F ∨ T) ∧ ~ F
Because of the order of operations, we will interpret ~ F first. This is interpreted as not false, which indicates that it is true.
( F ∨ T) ∧ ~ F
⇕
( F ∨ T) ∧ T
Let's interpret what's in the parentheses, ( F ∨ T). The disjunction of the statements is interpreted as F or T. The statement is true only when at least one of the components is true. T is true, so ( F ∨ T) is true.
( F ∨ T) ∧ T
⇕
T ∧ T
The conjunction of the statements is interpreted as T and T. The statement is true only when T and T are both true. T is true, so this statement is true. The statement in option B is the only false statement, and is therefore the correct answer.
