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Boolean Algebra: Foundations & Applications

Abstract

Boolean algebra underpins digital computing and programming through logical operations. This page outlines its historical evolution, laws, applications, and demonstrates its implementation using R.

Key Concepts

• AND (∧): Returns 1 if both inputs are 1
• OR (∨): Returns 1 if at least one input is 1
• NOT (¬): Inverts the input

R Code Demonstration

This code validates De Morgan's First Law using truth tables in R:

A <- c(0, 0, 1, 1) B <- c(0, 1, 0, 1) lhs <- ! (A & B) rhs <- (!A) | (!B) truth_table <- data.frame(A = A, B = B, `NOT(A AND B)` = lhs, `NOT A OR NOT B` = rhs) print(truth_table) cat("De Morgan's First Law holds:", all(lhs == rhs), "\n")

A B NOT(A AND B) NOT A OR NOT B
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0
De Morgan's First Law holds: TRUE

References

• Boole, G. (1847). The Mathematical Analysis of Logic. Macmillan.
• Boole, G. (1854). An Investigation of the Laws of Thought. Walton and Maberly.
• Hehner, E. (2012). “From Boolean algebra to unified algebra”. The Mathematical Intelligencer, 34(2), 3–11.
• Shannon, C. (1937). “A symbolic analysis of relay and switching circuits”. Transactions of the AIEE, 57(12), 713–723.