Question

Let proposition \(p\) be \(T,\) proposition \(q\) be \(F,\) and proposition \(r\) be \(T\). Find the truth values for the following: (a) \(p \vee q \vee r\) (b) \(p \vee(\neg q \wedge \neg r)\) (c) \(p \rightarrow(q \vee r)\) (d) \((q \wedge \neg p) \leftrightarrow r\) (e) \(\neg r \rightarrow(p \wedge q)\) (f) \((p \rightarrow q) \rightarrow \neg r\) (g) \(((p \wedge r) \rightarrow(\neg q \vee p)) \rightarrow(q \vee r)\)

Short answer

(a) T; (b) T; (c) T; (d) F; (e) T; (f) T; (g) T.

Step-by-step solution

Solve (a)

To find the truth value of \(p \vee q \vee r\), we need to evaluate the disjunction (OR operation) of \(p\), \(q\), and \(r\). Since \(p = T\), \(q = F\), and \(r = T\), the expression becomes \(T \vee F \vee T\).For disjunction, the result is true if at least one proposition is true. Hence, the truth value of \(p \vee q \vee r\) is \(T\).

Solve (b)

For \(p \vee (eg q \wedge eg r)\), we need to evaluate \(eg q\) and \(eg r\) first:- \(eg q\) is the negation of \(F\), which is \(T\).- \(eg r\) is the negation of \(T\), which is \(F\).Next, evaluate \(eg q \wedge eg r\), which is \(T \wedge F\). Using the AND operation, this evaluates to \(F\).The whole expression becomes \(p \vee F\), which is \(T \vee F\). This results in \(T\).

Solve (c)

For \(p \rightarrow (q \vee r)\), first evaluate \(q \vee r\):- Since \(q = F\) and \(r = T\), \(q \vee r = F \vee T = T\).Now evaluate \(p \rightarrow T\):- Since \(p = T\), \(T \rightarrow T\) is \(T\).Hence, the truth value of \(p \rightarrow (q \vee r)\) is \(T\).

Solve (d)

For \((q \wedge eg p) \leftrightarrow r\), let's find \(q \wedge eg p\):- \(eg p\) is \(F\) (since \(p = T\)).- \(q \wedge eg p = F \wedge F = F\).Now \(F \leftrightarrow T\):- The biconditional is true if both sides are the same. Since one side is \(F\) and the other is \(T\), \(F \leftrightarrow T = F\).So, the truth value is \(F\).

Solve (e)

For \(eg r \rightarrow (p \wedge q)\), start with:- \(eg r\) is \(F\).- \(p \wedge q = T \wedge F = F\) since for \(\wedge\) to be true both must be true.Now evaluate \(F \rightarrow F\):- In a conditional statement, if the premise is false, the whole statement is true. So \(F \rightarrow F = T\).Thus, the truth value is \(T\).

Solve (f)

For \((p \rightarrow q) \rightarrow eg r\), first find \(p \rightarrow q\):- \(p = T\), \(q = F\). So, \(T \rightarrow F = F\).Now the expression becomes \(F \rightarrow eg r\), where \(eg r = F\).- \(F \rightarrow F\) is \(T\) since a false antecedent makes a conditional statement true.Hence, the truth value is \(T\).

Solve (g)

For \(((p \wedge r) \rightarrow (eg q \vee p)) \rightarrow (q \vee r)\), start by solving \(p \wedge r\):- Since \(p = T\) and \(r = T\), \(p \wedge r = T\).Next, solve \(eg q \vee p\):- \(eg q = T\), so \(T \vee T = T\).Now \(T \rightarrow T = T\).Lastly, evaluate \(q \vee r\):- \(F \vee T = T\).So ultimately \(T \rightarrow T = T\).Thus, the truth value is \(T\).

Key concepts

Truth Tables

Truth tables are a powerful tool used in propositional logic to systematically explore the possible truth values of logical expressions. These tables help you understand how different input values affect the outcome of a logical statement. A truth table lists all possible combinations of truth values for given propositions and the resulting truth value of the expression. For example, if we have a simple expression like \(p \vee q\), where \(\vee\) represents the logical OR, a truth table would show every combination of \(p\) and \(q\) being true or false, and the corresponding truth value of the entire expression.
A truth table generally consists of the following:

• A column for each variable or proposition involved in the logical expression.
• A column for the expression itself, showing the result of the operation for each row.
• Rows corresponding to every possible combination of truth values of the propositions.

A truth table allows you to visualize and verify logical expressions, making it easier to draw conclusions or build upon more complex logical constructs.

Logical Connectives

Logical connectives are the building blocks of propositional logic. They are used to connect individual propositions to form more complex expressions. Understanding the different types of logical connectives is essential for creating and interpreting logical statements. The primary logical connectives include:

• **AND (\(\wedge\))**: Results in true if both propositions are true. Otherwise, it is false.
• **OR (\(\vee\))**: Results in true if at least one of the propositions is true. If both are false, the result is false.
• **NOT (\(eg\))**: Inverts the truth value of a single proposition. If the proposition is true, NOT makes it false and vice versa.
• **IMPLIES (\(\rightarrow\))**: Represents a conditional statement. It is false only when the first proposition (antecedent) is true, and the second (consequent) is false. Otherwise, it is true.
• **BICONDITIONAL (\(\leftrightarrow\))**: True if both propositions are either true or false. Therefore, it equates the two propositions' truth values.

Using these connectives allows you to build complex expressions and logically deduce the truth value of compositions of propositions.

Logical Equivalence

Logical equivalence is an important concept in propositional logic. It occurs when two logical statements have the same truth values for all possible input scenarios. In simpler terms, two expressions are logically equivalent if they always produce the same output no matter what the truth value of their propositions. Logical equivalence serves as a basis for simplifying logical expressions, which is helpful in mathematical proofs and computer algorithms. To determine if two expressions are logically equivalent:

• Create a truth table for each expression.
• Compare the resulting truth values in each table.

If the columns for both expressions match perfectly, the expressions are logically equivalent. For example, \(p \wedge q\) and \(q \wedge p\) are logically equivalent because they yield the same results regardless of the individual truth values of \(p\) and \(q\). Recognizing logical equivalence can simplify solutions and improve understanding of logical constructs.