Question

(a) Show that every formula containing only \(k\) (different) proposition letters is equivalent to a \(k-\mathrm{DNF}\) formula. (b) Show that \(p \leftrightarrow q\) is not equivalent to any 1 -DNF formula. (c) Show that for every natural number \(k\) (including 0 ), there is a formula containing only \(k+1\) (different) proposition letters that is not equivalent to any \(k-D N F\) formula.

Short answer

Every formula with k proposition letters can be a k-DNF; \(p \leftrightarrow q\) isn't a 1-DNF; larger k+1 letter formulas aren't k-DNF.

Step-by-step solution

Understanding k-DNF

A k-DNF (Disjunctive Normal Form) formula is a disjunction of conjunctions (ANDs of literals) with each conjunction containing at most k literals. For example, if k=2, a valid formula could be \((p \land q) \lor (r)\).

k-DNF Equivalence Proof

To show equivalence, convert any propositional formula with k proposition letters into a DNF. Consider truth tables: for each row where the formula is true, create a conjunction of literals matching the truth assignment for the proposition letters (negating if necessary). Take the disjunction of these conjunctions to form an equivalent k-DNF.

Non-Equivalence of biconditional and 1-DNF

The formula \(p \leftrightarrow q\) can be written as \((p \land q) \lor (eg p \land eg q)\). For a 1-DNF, each conjunction must only involve a single proposition. Neither \((p \land q)\) nor \((eg p \land eg q)\) can be reduced to a single literal without changing its truth values, proving 1-DNF isn't suitable.

Using Proposition Letters and 0-DNF

A formula containing only a single proposition letter (k+1 = 1) is not equivalent to a 0-DNF, since a 0-DNF can only be true for constant TRUE or FALSE, ignoring letter combinations.

Generalizing to k+1 Letters and k-DNF

For formula \(p_1 \land p_2 \land \ldots \land p_{k+1}\) (all conjunctions), there are \(2^{k+1}\) rows in the truth table. However, a k-DNF cannot handle all cases without covering all additional proposition cases, which violates the constraint of containing at most k proposition letters in each conjunction.

Key concepts

Propositional Logic

Propositional Logic is a branch of Discrete Mathematics that deals with propositions, which are statements that are either true or false. In this discipline, propositions are combined using logical connectives like AND (\(\land\)), OR (\(\lor\)), NOT (\(eg\)), and biconditional (\(\leftrightarrow\)). Propositional Logic forms the foundation for reasoning and decision-making processes in mathematics and computer science.

In practical terms, propositions are used to create logical formulas that can be evaluated as true or false. Understanding how to work with these formulas, including simplifying or finding equivalent expressions, is crucial in problem-solving. Each propositional formula can be assessed using a Truth Table—a structured way to see all possible true and false combinations.

By mastering Propositional Logic, you gain the ability to assess complex logical statements efficiently, an invaluable skill in fields such as software engineering, computational theory, and artificial intelligence.

Disjunctive Normal Form

Disjunctive Normal Form (DNF) is a specific way to express propositional logic formulas. In DNF, a formula consists of a disjunction (OR: \(\lor\)) of one or more conjunctions (AND: \(\land\)) of literals. A literal is either a proposition letter or its negation.

For example, the formula \((p \land q) \lor (r)\) is in DNF because it is a disjunction of the conjunction \((p \land q)\) and the literal \(r\). DNF is useful because it allows any propositional formula to be expressed in a standardized format, simplifying analysis and comparisons of logical expressions.

DNF is especially helpful in logic and circuit design because it provides a clear template for expressing every possible configuration of a set of conditions where the formula holds true. By converting to DNF, we can easily see the "true" states of logic circuits or constraints, which is a critical step in optimization and computation.

Truth Tables

Truth Tables are a fundamental tool in Propositional Logic used to determine the validity and equivalency of logical expressions. They consist of rows and columns where each column represents a proposition, and each row represents a possible truth assignment (true or false) for these propositions.

For every possible combination of truth values for a given set of propositions, a Truth Table shows the result of the formula. This is invaluable for systematically understanding how a logical statement behaves under different conditions. For instance, if a formula contains three propositions, the Truth Table will have \(2^3 = 8\) rows covering all combinations of truth assignments.

By constructing and examining Truth Tables, you can determine if two propositions are logically equivalent, identify tautologies (always true), contradictions (always false), or just render decision making and problem resolution straightforward and visual.

Logical Equivalence

Logical Equivalence refers to the scenario where two propositional formulas have the same truth values in all possible scenarios. This means, no matter what truth assignments are given to the proposition letters involved, both formulas will evaluate to true or false simultaneously.

Determining Logical Equivalence is crucial for simplifying complex logical expressions and making them more understandable or suitable for computation. For example, the formulas \(p \lor (q \land r)\) and \((p \lor q) \land (p \lor r)\) are logically equivalent due to their matching truth tables across all possible values of \(p\), \(q\), and \(r\).

To prove logical equivalence, constructing Truth Tables is a common method—if the final outcome column of both formulas matches perfectly, equivalence is established. This concept is widely employed in computer science, data management, law, and anywhere precise and consistent logic is required. Demonstrating logical equivalence allows for better optimization and clearer reasoning.