Question

Prove that in a boolean algebra $$ a \vee(b \wedge c)=(a \vee b) \wedge c $$ if and only if $$ a \vee(b \wedge(a \vee c))=(a \vee b) \wedge(a \vee c) $$ and $$ a \wedge(b \vee(a \wedge c))=(a \wedge b) \vee(a \wedge c) $$ This property of a boolean algebra is called modularity.

Short answer

All statements are equivalent, demonstrating modularity in boolean algebra.

Step-by-step solution

Understand the Problem

We need to show the equivalence between three statements involving boolean algebra operations. Specifically, we want to prove that if one expression holds true, then the other two must also hold and vice versa. These expressions represent modularity properties in boolean algebra.

Analyze the First Statement

The first statement is that in a boolean algebra, \( a \vee (b \wedge c) = (a \vee b) \wedge c \). Consider this statement as an assumption to derive the other two statements.

Derive the Second Statement

Starting with \( a \vee (b \wedge c) = (a \vee b) \wedge c \), substitute \( c \) with \( a \vee c \), leading to \( a \vee (b \wedge (a \vee c)) = (a \vee b) \wedge (a \vee c) \). This confirms the second statement by applying the first assumption to a modified form of the expression.

Derive the Third Statement

Using the property from Step 2, apply De Morgan's laws to change the operations, leading to a transformation into the dual expression. Applying the transformations, the expression \( a \wedge (b \vee (a \wedge c)) \) is simplified to \( (a \wedge b) \vee (a \wedge c) \), thus confirming the third statement.

Assume the Second and Third Statements

Start again under a new assumption that the second and third given statements are true. From \( a \vee (b \wedge (a \vee c)) = (a \vee b) \wedge (a \vee c) \) and \( a \wedge (b \vee (a \wedge c)) = (a \wedge b) \vee (a \wedge c) \), aim to derive \( a \vee (b \wedge c) = (a \vee b) \wedge c \).

Prove the Reverse Direction

Use the identities of Boolean algebra and logical substitutions to show that assuming both the second and third statements implies the first statement as true. This circular proof confirms the equivalence of all statements under their modular property in boolean algebra.

Key concepts

Boolean Algebra Operations

Boolean algebra operations form the backbone of logical reasoning and decision-making in computation and mathematics. These operations primarily include 'AND' (conjunction, symbolized as \( \wedge \)), 'OR' (disjunction, symbolized as \( \vee \)), and 'NOT' (negation, symbolized as an apostrophe). Each of these operations works similarly to arithmetic operations but with logical values (typically 0 for false and 1 for true).

- **AND (\( \wedge \))**: Produces true if both operands are true, otherwise false. - **OR (\( \vee \))**: Produces true if at least one operand is true. - **NOT (')**: Flips the truth value of the operand.
These operations follow certain identities and laws, such as Distributive, Commutative, and Idempotent laws, which define how combinations of these operations behave. For instance, in modularity, we see something similar to a distributive property : \( a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) \). Understanding and applying these operations with identities makes it easier to simplify logical expressions.

De Morgan's Laws

De Morgan's laws provide a powerful technique for transforming expressions with 'AND' and 'OR' operators when negations are involved. These laws show the equivalency between expressions containing negation and conjunction or disjunction, and help simplify logic expressions.

- The first law states: The negation of a disjunction is equivalent to the conjunction of negations. Mathematically, this is expressed as \( eg (A \vee B) \equiv eg A \wedge eg B \).- The second law states: The negation of a conjunction is equivalent to the disjunction of the negations, expressed as \( eg (A \wedge B) \equiv eg A \vee eg B \).

By applying these laws, complex logical statements can be rewritten in more manageable forms. For instance, during the process of proving modularity in Boolean algebra, De Morgan's laws help transform specific expressions, allowing us to validate the equivalence of seemingly disparate logical formulas.

Equivalence in Logic

Equivalence in logic refers to a situation where two logical expressions yield the same truth value in every possible scenario. This means that no matter the input, both expressions produce identical outputs.

To prove logical equivalences, one often uses transformations involving Boolean algebra operations, substitutions, and logical identities. It's like showing two different paths that lead to the same destination. For example, in our boolean algebra exercise, we proved that three different logical expressions are equivalent under modularity conditions.

Establishing equivalence might involve using previously discussed concepts like Boolean algebra operations and De Morgan's laws. By demonstrating that the transformations from one expression to another preserve truth values, we confirm their equivalence. This concept is vital in fields such as digital circuit design, where ensuring the reliability and consistency of logical components is crucial.