Question

Create truth tables for the following expressions. Include all necessary negations and sub-expressions.

1.( p ∧ q ) ∨ ( ¬ r )
2.¬ ( p ∨ q ) ∧ ( r ⊕ s )
3.[ ( q ⊕ r ) ∧ s ] ⟶ p
If you could explain the thought process behind how you did these that would be amazing, I am trying to learn the reasoning as I practice.

Answer 1

To create truth tables for the given expressions, evaluate the truth values for each variable and sub-expression. Break down each expression and explain the evaluation process.

Step-by-step explanation:

To create truth tables for the given expressions, we need to evaluate the truth values for each variable and sub-expression. Let's break down each expression:

1. ( p ∧ q ) ∨ ( ¬ r )
   • The sub-expression ( p ∧ q ) evaluates to true only when both p and q are true.
   • The sub-expression ( ¬ r ) negates the truth value of r.
   • The final expression evaluates to true if either of the sub-expressions is true.
2. ¬ ( p ∨ q ) ∧ ( r ⊕ s )
   • The sub-expression ¬ ( p ∨ q ) negates the truth value of the statement ( p ∨ q ).
   • The sub-expression ( r ⊕ s ) evaluates to true only when either r or s is true, but not both.
   • The final expression evaluates to true only when both sub-expressions are true.
3. [ ( q ⊕ r ) ∧ s ] ⟶ p
   • The sub-expression ( q ⊕ r ) evaluates to true only when either q or r is true, but not both.
   • The sub-expression ( [ ( q ⊕ r ) ∧ s ] ⟶ p ) evaluates to true if both ( q ⊕ r ) ∧ s and p are true.

By constructing truth tables, we can systematically evaluate all possible truth value combinations for the variables and sub-expressions, and determine the resulting truth values of the expressions.