Question

If p, q and r are three propositions, then which of the following combination of truth values of p, q and r makes the logical expression \({(p∨q)∧((∼p)∨r)}🡢((∼q)∨r)\) false?

• A. p = F, q = T, r = F
• B. p = T, q = F, r = T
• C. p = T, q = T, r = F
• D. p = T, q = F, r = F

Answer 1

The Correct Option is A

 To solve this problem, let's analyze the given logical expression and determine which combination of truth values makes it false. The logical expression is: \({(p∨q)∧((∼p)∨r)}🡢((∼q)∨r)\).

First, let's break down the logical expression:

1. The expression can be divided into two parts: the antecedent \((p∨q)∧((∼p)∨r)\) and the consequent 🡢), which is only false when the antecedent is true and the consequent is false.

Let's evaluate each option to find when this condition is met.

1. Option 1: p = F, q = T, r = F
   • Antecedent: 
     \({(p∨q)∧((∼p)∨r)}\)
     = \({(F∨T)∧((∼F)∨F)}\)
     = \({T∧T}\)
     = T
   • Consequent: \({((∼q)∨r)}\)
     = \({(∼T∨F)}\)
     = \({F∨F}\)
     = F
   • The implication \(T🡢F\) is false.
   • This combination makes the logical expression false.
2. Option 2: p = T, q = F, r = T
   • Antecedent: 
     \({(p∨q)∧((∼p)∨r)}\)
     = \({(T∨F)∧((∼T)∨T)}\)
     = \({T∧T}\)
     = T
   • Consequent: \({((∼q)∨r)}\)
     = \({(∼F∨T)}\)
     = \({T∨T}\)
     = T
   • The implication \(T🡢T\) is true.
3. Option 3: p = T, q = T, r = F
   • Antecedent: 
     \({(p∨q)∧((∼p)∨r)}\)
     = \({(T∨T)∧((∼T)∨F)}\)
     = \({T∧F}\)
     = F
   • Consequent: \({((∼q)∨r)}\)
     = \({(∼T∨F)}\)
     = \({F∨F}\)
     = F
   • The implication \(F🡢F\) is true.
4. Option 4: p = T, q = F, r = F
   • Antecedent: 
     \({(p∨q)∧((∼p)∨r)}\)
     = \({(T∨F)∧((∼T)∨F)}\)
     = \({T∧F}\)
     = F
   • Consequent: \({((∼q)∨r)}\)
     = \({(∼F∨F)}\)
     = \({T∨F}\)
     = T
   • The implication \(F🡢T\) is true.

The correct combination that makes the expression false is Option 1: p = F, q = T, r = F.