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Digital Circuits - Shift Registers

We know that one flip-flop can store one-bit of information. In order to store multiple bits of information, we require multiple flip-flops. The group of flip-flops, which are used to hold (store) the binary data is known as register . If the register is capable of shifting bits either towards right hand side or towards left hand side is known as shift register . An ‘N’ bit shift register contains ‘N’ flip-flops. Following are the four types of shift registers based on applying inputs and accessing of outputs. Serial In − Serial Out shift register Serial In − Parallel Out shift register Parallel In − Serial Out shift register Parallel In − Parallel Out shift register Serial In − Serial Out (SISO) Shift Register The shift register, which allows serial input and produces serial output is known as Serial In – Serial Out (SISO) shift register. The block diagram of 3-bit SISO shift register is shown in the following figure. This block d...

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: Venn Diagrams

Venn Diagrams Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets. Examples Set Operations Set Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product. Set Union The union of sets A and B (denoted by A ∪ B A ∪ B ) is the set of elements which are in A, in B, or in both A and B. Hence, A ∪ B = { x | x ∈ A O R x ∈ B } A ∪ B = { x | x ∈ A O R x ∈ B } . Example − If A = { 10 , 11 , 12 , 13 } A = { 10 , 11 , 12 , 13 } and B = { 13 , 14 , 15 } { 13 , 14 , 15 } , then A ∪ B = { 10 , 11 , 12 , 13 , 14 , 15 } A ∪ B = { 10 , 11 , 12 , 13 , 14 , 15 } . (The common element occurs only once) Set Intersection The intersection of sets A and B (denoted by A ∩ B A ∩ B ) is the set of elements which are in both A and B. Hence, A ∩ B = { x | x ∈ A A N D...

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