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Digital Circuits - Decoders

Decoder is a combinational circuit that has ‘n’ input lines and maximum of 2 n output lines. One of these outputs will be active High based on the combination of inputs present, when the decoder is enabled. That means decoder detects a particular code. The outputs of the decoder are nothing but the min terms of ‘n’ input variables (lines), when it is enabled. 2 to 4 Decoder Let 2 to 4 Decoder has two inputs A 1 & A 0 and four outputs Y 3 , Y 2 , Y 1 & Y 0 . The block diagram of 2 to 4 decoder is shown in the following figure. One of these four outputs will be ‘1’ for each combination of inputs when enable, E is ‘1’. The Truth table of 2 to 4 decoder is shown below. Enable Inputs Outputs E A 1 A 0 Y 3 Y 2 Y 1 Y 0 0 x x 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 From Truth table, we can write the Boolean functions for each output as Y 3 = E . A 1 . A 0 Y 3 = E . A ...

Discrete Mathematics - Rules of Inference

To deduce new statements from the statements whose truth that we already know, Rules of Inference are used. What are Rules of Inference for? Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements. An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “ ∴ ∴ ”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have. Table of Rules of Inference Rule of Inference Name Rule of Inference Name P ∴ P ∨ Q P ∴ P ∨ Q Addition P ∨ Q ¬ P ∴ Q P ∨ Q ¬ P ∴ Q Disjunctive Syllogism P Q ∴ P ∧ Q P Q ∴ P ∧ Q Conjunction P → Q Q → R ∴ P → R P → Q Q → R ∴ P → R ...

discrete mathematics:Introduction to Trees

Tree is a discrete structure that represents hierarchical relationships between individual elements or nodes. A tree in which a parent has no more than two children is called a binary tree. Tree and its Properties Definition − A Tree is a connected acyclic undirected graph. There is a unique path between every pair of vertices in G G . A tree with N number of vertices contains ( N − 1 ) ( N − 1 ) number of edges. The vertex which is of 0 degree is called root of the tree. The vertex which is of 1 degree is called leaf node of the tree and the degree of an internal node is at least 2. Example − The following is an example of a tree − Centers and Bi-Centers of a Tree The center of a tree is a vertex with minimal eccentricity. The eccentricity of a vertex X X in a tree G G is the maximum distance between the vertex X X and any other vertex of the tree. The maximum eccentricity is the tree diameter. If a ...

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