Problem 14 - Analysis with an introduction to proof

Question

Use truth tables to verify that each of the following is a tautology. Parts (a) and (b) are called commutative laws, parts (c) and (d) are associative laws, and parts (e) and (f) are distributive laws. (a) \((p \wedge q) \Leftrightarrow(q \wedge p)\) (b) \((p \vee q) \Leftrightarrow(q \vee p)\) (c) \([p \wedge(q \wedge r)] \Leftrightarrow[(p \wedge q) \wedge r]\) (d) \([p \vee(q \vee r)] \Leftrightarrow[(p \vee q) \vee r]\) (e) \([p \wedge(q \vee r)] \Leftrightarrow[(p \wedge q) \vee(p \wedge r)]\) (f) \([p \vee(q \wedge r)] \Leftrightarrow[(p \vee q) \wedge(p \vee r)]\)

Step-by-Step Solution

Answer

Through the construction and evaluation of truth tables for each expression, we have verified that all given expressions are tautologies. These include commutative laws (a) and (b), associative laws (c) and (d), and distributive laws (e) and (f).

Step 1: (a) Commutative law: Conjunction

Construct a truth table for the expression \((p \wedge q) \Leftrightarrow (q \wedge p)\): 1. Enumerate all possible values of \(p\) and \(q\): There are 2 possible values for each variable, yielding 4 total possibilities: \((T, T), (T, F), (F, T), (F, F)\). 2. Calculate \((p \wedge q)\) and \((q \wedge p)\) for each combination. 3. Determine whether \((p \wedge q) \Leftrightarrow (q \wedge p)\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Step 2: (b) Commutative law: Disjunction

Construct a truth table for the expression \((p \vee q) \Leftrightarrow (q \vee p)\): 1. Enumerate all possible values of \(p\) and \(q\): There are 2 possible values for each variable, yielding 4 total possibilities: \((T, T), (T, F), (F, T), (F, F)\). 2. Calculate \((p \vee q)\) and \((q \vee p)\) for each combination. 3. Determine whether \((p \vee q) \Leftrightarrow (q \vee p)\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Step 3: (c) Associative law: Conjunction

Construct a truth table for the expression \([p \wedge (q \wedge r)] \Leftrightarrow [(p \wedge q) \wedge r]\): 1. Enumerate all possible values of \(p\), \(q\), and \(r\): There are 2 possible values for each variable, yielding 8 total possibilities. 2. Calculate \([p \wedge (q \wedge r)]\) and \([(p \wedge q) \wedge r]\) for each combination. 3. Determine whether \([p \wedge (q \wedge r)] \Leftrightarrow [(p \wedge q) \wedge r]\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Step 4: (d) Associative law: Disjunction

Construct a truth table for the expression \([p \vee (q \vee r)] \Leftrightarrow [(p \vee q) \vee r]\): 1. Enumerate all possible values of \(p\), \(q\), and \(r\): There are 2 possible values for each variable, yielding 8 total possibilities. 2. Calculate \([p \vee (q \vee r)]\) and \([(p \vee q) \vee r]\) for each combination. 3. Determine whether \([p \vee (q \vee r)] \Leftrightarrow [(p \vee q) \vee r]\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Step 5: (e) Distributive law: Conjunction over Disjunction

Construct a truth table for the expression \([p \wedge (q \vee r)] \Leftrightarrow [(p \wedge q) \vee (p \wedge r)]\): 1. Enumerate all possible values of \(p\), \(q\), and \(r\): There are 2 possible values for each variable, yielding 8 total possibilities. 2. Calculate \([p \wedge (q \vee r)]\) and \([(p \wedge q) \vee (p \wedge r)]\) for each combination. 3. Determine whether \([p \wedge (q \vee r)] \Leftrightarrow [(p \wedge q) \vee (p \wedge r)]\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Step 6: (f) Distributive law: Disjunction over Conjunction

Construct a truth table for the expression \([p \vee (q \wedge r)] \Leftrightarrow [(p \vee q) \wedge (p \vee r)]\): 1. Enumerate all possible values of \(p\), \(q\), and \(r\): There are 2 possible values for each variable, yielding 8 total possibilities. 2. Calculate \([p \vee (q \wedge r)]\) and \([(p \vee q) \wedge (p \vee r)]\) for each combination. 3. Determine whether \([p \vee (q \wedge r)] \Leftrightarrow [(p \vee q) \wedge (p \vee r)]\) is true for all combinations. If the output is true for all combinations, then the expression is a tautology.

Key Concepts

Commutative Laws in Logic

In the realm of logical expressions, the commutative laws act as fundamental rules that denote how the order of terms does not affect the outcome of certain logical operations. Specifically, for any two propositions, say \(p\) and \(q\), the commutative laws can be expressed by the equivalences \( p \wedge q \Leftrightarrow q \wedge p \) for the logical conjunction (AND), and \( p \vee q \Leftrightarrow q \vee p \) for the logical disjunction (OR).

Understanding these laws is crucial for students as they lay the groundwork for constructing and simplifying complex logical propositions. Truth tables serve as a valuable tool for verifying these laws. By listing all possible truth values of the individual propositions and then calculating the resulting truth values for both sides of the equivalence, one can easily observe that the truth values match regardless of the order of the propositions. This affirms the nature of commutative laws as tautologies, meaning they are true in every possible scenario. Such properties make the study of logical statements more predictable and structured.

Associative Laws in Logic

When dealing with the combination of more than two propositions in logical operations, the associative laws provide essential guidance. These laws declare that the way in which propositions are grouped within parentheses does not affect the outcome for both conjunctions and disjunctions. The associative laws are formally defined by the equivalences \( [p \wedge (q \wedge r)] \Leftrightarrow [(p \wedge q) \wedge r] \) for conjunctions, and \( [p \vee (q \vee r)] \Leftrightarrow [(p \vee q) \vee r] \) for disjunctions.

By creating truth tables which cover all possible truth combinations for \(p\), \(q\), and \(r\), students can observe that the results on either side of the equivalences remain consistent. This consistency reinforces that the associative laws hold true, ensuring that no matter how propositions are nested or grouped, the logical outcome is invariant. Recognizing the associative laws helps students in simplifying complex logical expressions by rearranging the components without altering the truth value of the expressions.

Distributive Laws in Logic

The distributive laws offer yet another layer of structural regularity in logical expressions. These laws illustrate how a single proposition can be distributed across another combining operation and still produce an equivalent logical statement. The distributive laws can be showcased by the equivalences \( [p \wedge (q \vee r)] \Leftrightarrow [(p \wedge q) \vee (p \wedge r)] \) and \( [p \vee (q \wedge r)] \Leftrightarrow [(p \vee q) \wedge (p \vee r)] \) for the combination of conjunction and disjunction.

Using truth tables, students can confirm that the logical equivalence expressed by the distributive laws consistently holds true across all permutations of truth values for the involved propositions. This verification process not only solidifies understanding but also empowers students to manipulate and break down complex logical statements into more understandable parts, or to combine simple statements into a more compact form. The distributive laws are essential when it comes to optimizing logical expressions and making them more tractable for analysis or implementation in digital logic circuits, for example.