Boolean Algebra and Digital Logic Circuits

Vocabulary flashcards covering core concepts of Boolean algebra, logic gates, minimization methods, combinational circuits, and sequential circuits based on the lecture transcript.

Boolean Algebra

A system of mathematical logic that performs arithmetical and logical operations on digital signals represented by binary values $0$ and $1$.

Literals

The variables and their complements present in a Boolean expression.

Duality Principle

A rule stating that a Boolean expression remains valid if OR ($+$) and AND ($\cdot$) operations are interchanged, and constants$0$and$1$$ are interchanged, while variables and complements remain unchanged.

Absorption Law

A Boolean theorem stating that $A + AB = A$ and $A(A + B) = A$.

De Morgan's First Law

A Boolean law stating that $\\overline{A + B} = \\bar{A} \\cdot \\bar{B}$, proving that a NOR gate is equivalent to a bubbled AND gate (negative AND gate).

De Morgan's Second Law

A Boolean law stating that $\\overline{A \\cdot B} = \\bar{A} + \\bar{B}$, proving that a NAND gate is equivalent to a bubbled OR gate (negative OR gate).

Universal Gate

A logic gate, such as NAND or NOR, that can be used to implement any Boolean logic operation without requiring any other type of gate.

XOR Gate

A logic gate with output $Y = A \\oplus B = \\bar{A}B + A\\bar{B}$, which outputs logic $1$ when the two inputs are different.

XNOR Gate

A logic gate with output $Y = A \\odot B = AB + \\bar{A}\\bar{B}$, which outputs logic $1$ when the two inputs are identical.

Minterm

A product term in standard Sum of Products (SOP) form containing every variable in either complemented or uncomplemented form, evaluating to $1$ for a specific combination of input values.

Maxterm

A sum term in standard Product of Sums (POS) form containing every variable in either complemented or uncomplemented form, evaluating to $0$ for a specific combination of input values.

Karnaugh Map (K-map)

A visual diagram tool composed of $2^n$ cells used to minimize and simplify Boolean logic expressions for $2$, $3$, $4$, $5$, and $6$ variables.

Don't Care Condition

An input condition in a logic circuit that never occurs or produces an undefined output level, which can be treated as either $0$ or $1$ (marked with 'X') to optimize logic simplification.

Quine-McCluskey Method

A tabular algorithmic method used to simplify Boolean expressions, particularly useful when dealing with a large number of variables.

Combinational Circuit

A logic circuit whose output depends exclusively on the present input values and contains no memory or storage elements.

Half Adder

A combinational circuit that adds two 1-bit binary inputs, producing a sum output ($S = A \\oplus B$) and a carry-out output ($\\text{Cout} = AB$).

Full Adder

A combinational circuit that adds three 1-bit inputs ($A$, $B$, and $C_{in}$), producing a sum output ($S = A \\oplus B \\oplus C_{in}$) and a carry-out output ($\\text{Cout} = AB + B C_{in} + A C_{in}$).

Half Subtractor

A combinational circuit that subtracts one 1-bit input from another, producing a difference output ($\\text{Diff} = A \\oplus B$) and a borrow-out output ($\\text{Borrow} = \\bar{A}B$).

Full Subtractor

A combinational circuit that subtracts two bits ($A$ and $B$) with a borrow-in ($B_{in}$), producing a difference output ($\\text{Diff} = A \\oplus B \\oplus B_{in}$) and a borrow-out output ($\\text{Bout} = \\bar{A}B + B_{in}(\\overline{A \\oplus B})$).

Multiplexer (MUX)

A combinational circuit (also known as a data selector or many-to-one circuit) that selects one of $2^n$ input lines using $n$ select lines and routes it to a single output line.

Demultiplexer (DEMUX)

A combinational circuit (one-to-many circuit) that directs a single input line to one of $2^n$ output lines based on $n$ select lines.

Decoder

A combinational logic circuit that converts coded digital information from $n$ input lines into up to $2^n$ unique output lines.

Encoder

A combinational logic circuit that performs the inverse operation of a decoder by converting up to $2^n$ input lines into an $n$-bit coded output.

Sequential Circuit

A logic circuit whose output depends on both the present inputs and the past inputs stored in memory elements (present state).

Synchronous Sequential Circuit

A sequential circuit whose state transitions and operations are synchronized by a periodic timing signal known as a clock signal ($clk$).