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Questions and Answers
Who proposed algebra for symbolically representing problems in logic?
Who proposed algebra for symbolically representing problems in logic?
George Boole
What are the mathematical systems founded upon the work of Boole called?
What are the mathematical systems founded upon the work of Boole called?
Boolean algebra
Who introduced the application of Boolean algebra to certain engineering problems?
Who introduced the application of Boolean algebra to certain engineering problems?
C.E. Shannon
The formal definition of Boolean algebra employs the postulates formulated by ______ in 1904.
The formal definition of Boolean algebra employs the postulates formulated by ______ in 1904.
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What forms the basic assumption of a mathematical system?
What forms the basic assumption of a mathematical system?
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What are the most common postulates used to formulate various structures?
What are the most common postulates used to formulate various structures?
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A set S is closed w.r.t. a binary operator if for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S.
A set S is closed w.r.t. a binary operator if for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S.
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The result of each operation with operator (+) or (.) is neither 1 or 0.
The result of each operation with operator (+) or (.) is neither 1 or 0.
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A set S is said to have an identity element w.r.t a binary operation * on S, if there exists an element e ∈ S with the property, e* x = x * e = x
A set S is said to have an identity element w.r.t a binary operation * on S, if there exists an element e ∈ S with the property, e* x = x * e = x
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A binary operator * on a set S is said to be commutative if, xy=yx for all x, y ∈ S
A binary operator * on a set S is said to be commutative if, xy=yx for all x, y ∈ S
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If * and • are two binary operation on a set S, • is said to be distributive over + whenever, x . (y+ z) = (x. y) + (x. z)
If * and • are two binary operation on a set S, • is said to be distributive over + whenever, x . (y+ z) = (x. y) + (x. z)
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A set S having the identity element e, w.r.t. binary operator * is said to have an inverse, whenever for every x∈ S, there exists an element x’∈ S such that, x.x’∈ e
A set S having the identity element e, w.r.t. binary operator * is said to have an inverse, whenever for every x∈ S, there exists an element x’∈ S such that, x.x’∈ e
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X+x=x is a property called Absorption Theorem
X+x=x is a property called Absorption Theorem
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X.x=x is a property called Absorption Theorem
X.x=x is a property called Absorption Theorem
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X + 0 = x is a property called Identity Element
X + 0 = x is a property called Identity Element
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X . 1 = x is a property called Identity Element
X . 1 = x is a property called Identity Element
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X+1=1 is a property called Associative Property
X+1=1 is a property called Associative Property
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X (y+z) = xy+ xz is a property called Distributive Property
X (y+z) = xy+ xz is a property called Distributive Property
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X + x’ = 1 is a property called Inverse
X + x’ = 1 is a property called Inverse
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X.x’ = 0 is a property called Inverse
X.x’ = 0 is a property called Inverse
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Boolean addition is commutative.
Boolean addition is commutative.
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Boolean multiplication is commutative.
Boolean multiplication is commutative.
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Boolean addition is associative.
Boolean addition is associative.
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Boolean multiplication is associative.
Boolean multiplication is associative.
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(AB)’= A’ + B’
(AB)’= A’ + B’
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(A+B)’ = A’.B’
(A+B)’ = A’.B’
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Boolean algebra is not dual.
Boolean algebra is not dual.
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The consensus theorem is a method for simplifying Boolean expressions.
The consensus theorem is a method for simplifying Boolean expressions.
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A minterm results from an AND operation of variables.
A minterm results from an AND operation of variables.
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A maxterm results from an AND operation of variables.
A maxterm results from an AND operation of variables.
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In a standard sum-of-products expression, all literals are present.
In a standard sum-of-products expression, all literals are present.
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In a standard product-of-sums expression, not all literals are present.
In a standard product-of-sums expression, not all literals are present.
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A Karnaugh map can be used to simplify Boolean expressions.
A Karnaugh map can be used to simplify Boolean expressions.
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The number of cells in a Karnaugh map is equal to 2n where n is the number of variables?
The number of cells in a Karnaugh map is equal to 2n where n is the number of variables?
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What is the minimum sum of products term for the following Boolean function: xy + x’z + yz?
What is the minimum sum of products term for the following Boolean function: xy + x’z + yz?
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What is the minimum sum of products term for the Boolean equation: (x+ y) (x’+ z) (y + z)?
What is the minimum sum of products term for the Boolean equation: (x+ y) (x’+ z) (y + z)?
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What is the minimum sum of products term for the Boolean equation: x’y + xy + x’y’?
What is the minimum sum of products term for the Boolean equation: x’y + xy + x’y’?
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What is the minimum sum of products term for the Boolean function: x + xy’ + x’y?
What is the minimum sum of products term for the Boolean function: x + xy’ + x’y?
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What is the minimum sum of products term for the Boolean equation: AB + (AC)'+ AB'C (AB + C)?
What is the minimum sum of products term for the Boolean equation: AB + (AC)'+ AB'C (AB + C)?
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What is the minimum sum of products term for the Boolean equation: x’y + xy + xyz?
What is the minimum sum of products term for the Boolean equation: x’y + xy + xyz?
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What is the minimum sum of products term for the Boolean equation: xyz + xy’z + xyz’?
What is the minimum sum of products term for the Boolean equation: xyz + xy’z + xyz’?
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What is the minimum sum of products term for the Boolean equation: x’y’z’ + x’yz’ + xy’z’ + xyz’?
What is the minimum sum of products term for the Boolean equation: x’y’z’ + x’yz’ + xy’z’ + xyz’?
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What is the minimum sum of products term for the Boolean equation: w’xyz’ + xyz’ + xy’z’ + xy’z?
What is the minimum sum of products term for the Boolean equation: w’xyz’ + xyz’ + xy’z’ + xy’z?
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What is the minimum sum of products term for the Boolean function: w’xy’z + w’xyz + wxz?
What is the minimum sum of products term for the Boolean function: w’xy’z + w’xyz + wxz?
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What is the minimum sum of products term for the Boolean equation: x’y’z’ + x’y’z + x’yz’ + x’yz + xy’z’?
What is the minimum sum of products term for the Boolean equation: x’y’z’ + x’y’z + x’yz’ + x’yz + xy’z’?
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What is the minimum sum of products term for the Boolean equation: w’y(w’xz)’ + w’xy’z’ + wx’y?
What is the minimum sum of products term for the Boolean equation: w’y(w’xz)’ + w’xy’z’ + wx’y?
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What is the minimum sum of products term for the Boolean equation: xy + x(y+z) + y(y+z)?
What is the minimum sum of products term for the Boolean equation: xy + x(y+z) + y(y+z)?
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What is the minimum sum of products term for the Boolean equation: xy’(z+wy) + x’y’] z?
What is the minimum sum of products term for the Boolean equation: xy’(z+wy) + x’y’] z?
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What is the minimum sum of products term for the Boolean equation: x’yz + xy’z’ + x’y’z’ + xy’z + xyz ?
What is the minimum sum of products term for the Boolean equation: x’yz + xy’z’ + x’y’z’ + xy’z + xyz ?
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What is the minimum sum of products term for the Boolean equation: [(xy)’ + x’ + xy]’?
What is the minimum sum of products term for the Boolean equation: [(xy)’ + x’ + xy]’?
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What is the minimum sum of products term for the Boolean equation: [xy + xz]’ + x’y’z?
What is the minimum sum of products term for the Boolean equation: [xy + xz]’ + x’y’z?
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What is the minimum sum of products term for the Boolean equation: xy + xy’(x’z’)’ ?
What is the minimum sum of products term for the Boolean equation: xy + xy’(x’z’)’ ?
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What is the minimum sum of products term for the Boolean expression: [(xy)’ + x’ + xy]’?
What is the minimum sum of products term for the Boolean expression: [(xy)’ + x’ + xy]’?
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What is the minimum sum of products term for the Boolean expression: [(xy+z’) ((x+y)’+z) ]’?
What is the minimum sum of products term for the Boolean expression: [(xy+z’) ((x+y)’+z) ]’?
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What is the minimum sum of products term for the Boolean expression: (x+y) (x’z’+z) (y’ + xz)’?
What is the minimum sum of products term for the Boolean expression: (x+y) (x’z’+z) (y’ + xz)’?
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What is the minimum sum of products term for the Boolean expression: x’y’z’ + x’yz’ + xy’z’ + xyz’?
What is the minimum sum of products term for the Boolean expression: x’y’z’ + x’yz’ + xy’z’ + xyz’?
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What is the minimum sum of products term for the Boolean function: xy’ (z + wy) + x’y’ ] z?
What is the minimum sum of products term for the Boolean function: xy’ (z + wy) + x’y’ ] z?
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What is the minimum sum of products term for the Boolean equation: x’yz + xy’z’ + x’y’z’ + xy’z + xyz?
What is the minimum sum of products term for the Boolean equation: x’yz + xy’z’ + x’y’z’ + xy’z + xyz?
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What is the minimum sum of products term for the Boolean equation: [(xy)’ + x’ + xy]’?
What is the minimum sum of products term for the Boolean equation: [(xy)’ + x’ + xy]’?
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What is the minimum sum of products term for the Boolean function: (xy + xz)’ + x’y’z?
What is the minimum sum of products term for the Boolean function: (xy + xz)’ + x’y’z?
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What is the minimum sum of products term for the Boolean equation: xy + xy’(x’z’)’ ?
What is the minimum sum of products term for the Boolean equation: xy + xy’(x’z’)’ ?
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What is the minimum sum of products term for the Boolean expression: [(xy’ + xyz)’ + x(y+xy’)]’?
What is the minimum sum of products term for the Boolean expression: [(xy’ + xyz)’ + x(y+xy’)]’?
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What is the minimum sum of products term for the Boolean expression: [(xy+z’) ((x+y)’+z) ]’?
What is the minimum sum of products term for the Boolean expression: [(xy+z’) ((x+y)’+z) ]’?
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What is the minimum sum of products term for the Boolean expression: (x+y) (x’z’+z) (y’ + xz)’?
What is the minimum sum of products term for the Boolean expression: (x+y) (x’z’+z) (y’ + xz)’?
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Study Notes
Electronic Circuits
- George Boole proposed algebra for symbolically representing logic problems in 1854.
- Boolean algebra is named after him.
- Boolean algebra was used in engineering problems in 1938 by C.E. Shannon.
- E.V. Huntington defined postulates for Boolean algebra in 1904.
Fundamental Postulates of Boolean Algebra
- Postulates are the basic assumptions from which theorems and properties are derived in a mathematical system.
- Closure: A set is closed with respect to a binary operator if every pair of elements in the set produces a unique element in the set under that operation. The result of each operation with the operator or is either 1 or 0.
- Identity Element: A set is said to have an identity element if there exists an element e within the set that satisfies the equation e * x = x * e = x for all x in the set.
- eg. 0 + 0 = 0; 0 + 1 = 1 + 0 = 1; 11 = 1, 10=0*1=1
- a*0= a
- a*1= a
- Commutative Law: A binary operator * on a set S is said to be commutative if x * y= y * x for all x, y ∈ S.
- eg. A + B = B + A; A * B = B * A
- Distributive Law: If * and • are two binary operations on a set S, • is said to be distributive over + whenever x • (y + z) = (x • y) + (x • z)
- eg. A * (B + C) = (A * B) + (A * C); A+ (B * C) = (A + B) * (A + C)
- Inverse: A set S having the identity element e, is said to have an inverse, whenever for every x ∈ S, there exists an element x' ∈ S such that x * x' = e
- eg. x + x' = 1; x * x' = 0
Theorems and Properties of Boolean Algebra
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Theorems are derived from postulates.
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x + x = x; x ∗ x = x
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x + 1 = 1; x ∗ 0 = 0
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(x')' = x
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x + xy = x; x (x+ y) = x
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x+x'y= x+ y; x(x'+ y)= xy
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Associative property: A + (B + C) = (A + B) + C and A * (B * C) = (A * B) * C
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Distributive property: A + BC = (A + B) (A + C) and A (B + C) = AB + AC
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Commutative property: A + B = B + A and A * B = B * A
Boolean Functions
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Boolean function simplification is done using Boolean algebra properties, laws and theorems. Examples:
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x(x'+y) = xy
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x + x'y = x + y
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(x+y)(x+y') = x
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xy + x'z + yz = xy + x'z
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xy + yZ + y'z = xy + z
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(x+y)(x'+z)(y+z) = xy + xyz + x'z + x'yz
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xy' + xy + x'y'= y+ x'y'
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x'y'z'+x'yz'+xy'z'+xyz'=z'(x'+x)+y'z'(x'+x) =z'+xy'
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w'xy'z+ w'xyz+ wxz = xz(w'+w) =xz
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