CLASS -11 COMPUTER SCIENCE
CHAPTER- 14 BOOLEAN LOGIC
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NOTES FOR COMPLETE UNDERSTANDING -
- Boolean Logic, also known as boolean algebra was developed by mathematician “George Boole”.
- Later, Boolean logic helped in solving many big problems.
- These logics also helped in developing circuits.
- Complete logic is based on Yes or No which is also represented as True(T) or False(F).
- Computer system transmits its signals on high voltage or low voltage which can be denoted by true or false and with which we can develop it’s circuit.
- Binary Values (Quantities) ;
- In our daily life, we need to take logical decisions where answers comes in either Yes (T) or No (F).
- For ex-
- Should I have tea?
- The decisions which results in Yes or No only are known as binary decisions.
- Values T and F are known as truth values.
- Boolean variables can only be either T or F.
- Indira Gandhi was the first lady Prime Minister of India (it results in true -T). Therefore it is a logic statement.
- What do you want to eat ?( it can not be result in T or F). Therefore it is not a logic statement.
- Logical Operations ;
- These are of three types - • AND • OR • NOT
- These are used to develop compound statement.
- For ex - • He prefers coffee not tea. • He plays guitar and sitar. • I watch TV on Sunday or I go for a movie.
- These are also used to join logical variables.
- For ex- X NOT Y OR Z , Y AND Z OR X
- Logical Operators (NOT) :
- NOT operator is represented by applying ( ‘ ) or ( ¯ ) on an expression.
- This is a unary operator also known as complementation operator .
- Ex - 0’ = 1 and 1’ = 0
- Truth table and Venn Diagram for NOT operator is as under :
- Logical Operators (AND) :
- AND operator is used by applying ( . ) between two variables. like X.Y
- It shows a very important operation of boolean algebra which is known as logical multiplication.
- X.Y is to be read as X AND Y.
- Rules of AND operation are ~ 0 . 0 = 0 ~0 . 1 = 0 ~1 . 0 = 0 ~1 . 1 = 1
- AND operator’s truth table and Venn Diagram are as under ;
- Logical Operators (OR) ;
- OR operator is used by applying ( + ) between two variables , like X+Y
- It shows a very important operation of boolean algebra which is known as logical addition.
- X+Y is to be read as X OR Y.
- Rules of OR operation are ~0 + 0 = 0 ~ 0 + 1 = 1 ~1 + 0 = 1 ~1 + 1 = 1
- OR operator’s truth table and Venn Diagram are as under :
- Evaluation of Boolean Expression using Truth Table :
- To develop Boolean Expression, logical variables are used with logical operators.
- Truth table is used for its evaluation.
- Basic Logic Gates :
- When Shannon used boolean logic in switching circuit of telephone then Engineers realized that boolean algebra can also be used in computer electronics.
- In computers, logic gates are used to complete boolean operations.
- A Gate is a basic Electronic Circuit used to develop an output signal by passing one or more input signal(s).
- A Gate is a kind of two-state digital circuit because its input or output are either low voltage (i.e. 0) or high voltage (i.e.1).
- Gates are also known as logic gates because they can coordinate with boolean logic.
- Logic gates are of three types-- 1. Inverter (NOT Gate) 2. OR Gate 3. AND Gate
- Inverter (NOT Gate) :
- NOT gate takes one input and produces one output.
- Its output is exactly opposite to its input.
- It is called NOT gate because its output is not similar to its input.
- Its output is also called as complement(opposite) of input.
- Truth table and diagram of NOT gate are as under :
- OR Gate :
- OR gate takes two or more inputs and produces one output.
- Its output is logical sum of passed inputs.
- It produces high voltage output on any one high voltage input. It produces low voltage output only when all the inputs are of low voltage.
- This gate work for more than two input signals in similar manner.
- AND Gate :
- AND gate takes two or more inputs and produces one output.
- Its output is logical multiplication of passed inputs.
- It produces low voltage output on any of the input signal as low voltage input. It produces high voltage output only when all the inputs are of high voltage.
- AND Gate Truth table and diagram are as under-
- This gate work for more than two input signals in similar manner.
- Boolean logic-Basic Postulates :
- Boolean algebra is a kind of mathematical system because of which it is based on some fundamental laws which are known as basic postulates.
- Which are as follows- -
- I. If X ≠ 0 then X = 1 and if X ≠ 1 then X = 0
- II. OR Relations (logical addition) ~ 0 + 0 = 0 ~ 0 + 1 = 1 ~ 1 + 0 = 1 ~ 1 + 1 = 1
- III. AND Relations (logical multiplication) ~ 0 . 0 = 0 ~ 0 . 1 = 0 ~ 1 . 0 = 0 ~ 1 . 1 = 1
- IV. Complementary Rules: ~ 0’ = 1 ~ 1’ = 0
- Principle of DUALITY :
- It is very important concept of boolean logic.
- A boolean relation can be drived by other boolean relation if we follow following rules --
- 1. Every OR signal (+) is replaced by AND signal (.).
- 2. Every AND signal (.) is replaced by OR signal (+).
- 3. Every 1 is replaced by 0 and every 0 is replaced by 1.
- The received expression will be dual expression of original expression.
- Main Concepts of Boolean Algebra :
- I. Properties of 0 and 1 --
- 0 + X = X
- 1 + X = 1
- 0 . X = 0
- 1 . X = X
- II. Idempotence Law - (a) X + X = X (b) X . X = X
- III. Involution – (X’)’ = X
- IV. Complementarity Law – (a) X + X’ = 1 (b) X . X’ = 0
- V. Commutative Law – (a) X + Y = Y + X (b) X . Y = X . Y
- VI. Associative Law – (a) X+(Y+Z) = (X+Y)+Z (b) X . (Y . Z) = (X . Y) . Z
- VII. Distributive Law – (a) X . (Y + Z) = X.Y + X.Z (b) X + Y.Z = (X + Y) . (X + Z)
- VIII. Absorption Law – (a) X + X . Y = X (b) X . (X + Y) = X
- IX. Other rules (Third Distributive Law) X + X’.Y = X + Y X
- X. DEMORGAN’s Law – (a) (X+Y)’ = X’ . Y’ (b) (X.Y)’ = X’ + Y’
- You can prove a law with the help of truth table and can also be proven mathematically.
- Minimization of Boolean Expression algebraically;
- Minimize the expression- AB’CD’ + AB’CD +ABCD’ + ABCD
- Solution: AB’CD’ + AB’CD +ABCD’ + ABCD = AB’C(D’+D)+ABC(D’+D) =AB’C + ABC = AC (B’+B) = AC.1 = AC
- Other Logic Gates :
- There are some derived gates also besides AND, OR, NOT Gates.
- Like- NOR, NAND, XOR, XNOR Gates.
- NOR Gate: There is two or more input signals and one output signal in NOR gate if all input signals are 0 (low) then output signal will be 1 (high).
- NAND Gate : There is two or more input signals and one output signal in NAND gate if all input signals are 1 (high) then output signal will be 0 (low).
- It is also known as universal gate because fundamental gates can be retrieved with its help.
- XOR Gate : XOR Gate produces 1 (high) when inputs are not at equal logic level.
- It is also known as universal gate because fundamental gates can be retrieved with its help.
- XNOR Gate : XNOR Gate produces 1 (high) when the inputs are identical (both 1’s or both 0’s).
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