Question

Prove that in a boolean algebra, DeMorgan's Laws hold; that is, $$ \begin{array}{l} \neg(x \vee y)=\neg x \wedge \neg y \\ \neg(x \wedge y)=\neg x \vee \neg y \end{array} $$

Short answer

DeMorgan's Laws are proven in Boolean algebra: \(\neg(x \vee y) = \neg x \wedge \neg y\) and \(\neg(x \wedge y) = \neg x \vee \neg y\).

Step-by-step solution

Understand Boolean Algebra Basics

Boolean algebra is a mathematical structure used to perform operations on logical values. The primary operations are logical AND (\( \wedge \)), logical OR (\( \vee \)), and logical NOT (\( eg \)). The elements of this algebra are typically TRUE (1) and FALSE (0), and the algebra obeys specific properties like commutativity, associativity, distributivity, identity elements, complementation, and DeMorgan's Laws.

State DeMorgan's Laws

DeMorgan's Laws in the context of Boolean algebra are stated as the following identities:\[ eg(x \vee y) = eg x \wedge eg y \] and \[ eg(x \wedge y) = eg x \vee eg y \]. These laws demonstrate the interaction between negation and the two binary operations: AND and OR.

Prove \(\neg(x \vee y) = \neg x \wedge \neg y\)

To prove this, consider the properties of complementation and distributivity. First, we use the property of complementation: for any element \(z\), \(eg(eg z) = z\). We can substitute \(eg x\) and \(eg y\) into the OR operation and apply the distributive laws and identities:\[ eg(x \vee y) = eg((x \vee y) \vee 0) = eg(x \vee y) \wedge 1 = eg x \wedge eg y.\] Thus, \(eg(x \vee y) = eg x \wedge eg y\) as required.

Prove \(\neg(x \wedge y) = \neg x \vee \neg y\)

Similarly, use the properties of complement and distributivity to prove this part. Begin by complementing the AND operation:\[ eg(x \wedge y) = eg((x \wedge y) \wedge 1) = eg(x \wedge y) \vee 0 = eg x \vee eg y.\] Therefore, \(eg(x \wedge y) = eg x \vee eg y\), validating the second part of DeMorgan's Laws.

Key concepts

Boolean Algebra

Boolean algebra is a branch of algebra that is primarily used to analyze and simplify digital circuits such as computer hardware. It operates on binary variables and logic operations, making it fundamental in designing circuits and logic-based systems. The basic components, often represented by 0 and 1, correspond to FALSE and TRUE in logical operations. In Boolean algebra, there are three primary operations:

• AND (\( \wedge \) ): A binary operation that requires both operands to be TRUE for the result to be TRUE.
• OR (\( \vee \) ): A binary operation where at least one operand must be TRUE for the result to be TRUE.
• NOT (\( eg \) ): A unary operation that inverts the truth value of the operand, turning TRUE to FALSE and vice versa.

Boolean algebra follows specific rules and laws such as commutativity, associativity, and distributivity, which make it easier to work with complex expressions. At the heart of these rules are identity and null elements that make simplifying expressions possible without changing their truth value.

Logical Operations

Logical operations are the backbone of Boolean algebra and involve manipulating variables and expressions to obtain a result in a binary format. These operations are crucial for evaluating expressions within computer science, engineering, and mathematics.
The logical AND, OR, and NOT are the foundational operations that allow the construction of complex expressions by combining multiple variables.

• AND in logical operations (\( \wedge \) ) is akin to multiplication, as both operands must be `1` (TRUE) for the result to be `1` (TRUE).
• OR (\( \vee \) ) is similar to addition because at least one operand needs to be `1` (TRUE) for the result to be `1` (TRUE).
• NOT (\( eg \) ) flips the value; `1` becomes `0` (FALSE) and vice versa.

Logical operations form the basis for constructing truth tables, which visually represent how inputs affect outputs through various logical gates and operators in digital systems.

Proof Techniques

Proof techniques are critical in validating theorems and properties in mathematics, including Boolean algebra. They offer a systematic approach to demonstrate that certain statements or identities, such as DeMorgan's Laws, are universally true.
In Boolean algebra, proving identities often involves logical reasoning along with properties of operations:

• Substitution: Replace variables or expressions with equivalent counterparts to simplify the main expression.
• Use of Properties: Engage properties such as associativity, distributivity, and complementation to transform expressions.
• Truth Tables: Evaluate expressions across all possible truth values of the variables to verify consistency.

For example, to prove DeMorgan's Laws, one might substitute expressions and apply complement and distributive properties to simplify and re-arrange terms to reach the conclusion. Such methods build the foundation for more advanced mathematical reasoning and problem-solving.