11.1 Boolean Functions

Boolean Algebra• An algebra is a set with one or more operations defined

on it.• A boolean algebra has three main operations, and, or, and

not, (typically operating on the set {0,1}).

• Rules of precedence for Boolean operators: PCoPS- parentheses, complements, products, sums

OR + a+b

AND ∙ ab

NOT complement

Examples:

1 ∙0+1 ∙ (1+1 )+0=¿

(1 ∙1)+0 ∙1=¿

Laws of Boolean Algebra

Double Complement

Idempotent

Identity

Domination

00

11

1

0

x

x

xx

xx

xxx

xxx

xx

More Laws of Boolean Algebra

Commutative

Associative

Distributive

))((

)(

)()(

)()(

zxyxyzx

xzxyzyx

zxyyzx

zyxzyx

xyyx

xyyx

Laws of Boolean Algebra Concluded

DeMorgan’s Laws

Absorption Laws

Unit Property

Zero Property 0

1

)(

xx

xx

xyxx

xxyx

yxyx

yxxy

“Proving” the Laws of Boolean Algebra

• If the underlying set is just {0,1}, we can prove these laws with truth tables, just as we did with propositional logic.

Duals and the Duality Principle

• The dual of a boolean expression is obtained by replacing all sums by products, all products by sums, all 0’s by 1’s, and all 1’s by 0’s.

• Example:

• The duality principle says that if an equation in a boolean algebra is an identity, i.e. always true no matter what the values of the variables, then the equation obtained by replacing both sides by their duals is also an identity.

Boolean Functions of Degree n on the Boolean algebra {0, 1}

𝐹 (𝑥 , 𝑦 ,𝑧 )=𝑥 𝑦+𝑦 𝑧

Example: Boolean Function Table

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

0 0 0

Using a 3-cube to represent a Boolean Function:

Lattices and Boolean Algebra

• A lattice is a partially ordered set in which every pair of elements has both a least upper bound (lub) and a greatest lower bound (glb).

• The supremum ab is defined as lub(a,b) and the infimum ab is defined as glb(a,b)

• A lattice is said to be distributive if each of and distributes over the other.

• For a distributive lattice to be a Boolean algebra, there must be (a) a largest element 1, (b) a least element 0, and (c) for each element x a complement with the property that

x0 and 1 xxxx

11.2 Representing Boolean Functions

( n-ary functions on the set {0, 1} )

Disjunctive Normal Form

• Pick out all the ones in the “function table” column corresponding to the function value.

• Translate each to an “and” of n literals (a literal is a Boolean variable of the form x or )

• Each such product is called a “minterm”.• The desired function is the sum of these

minterms, and is called the sum-of-products expansion or disjunctive normal form of the function.

x

Example

1 1

1 0

0 1

0 0

𝐹 (𝑥 , 𝑦 )=𝑥+𝑦+(𝑥+𝑦 )

Another Example

• Find the DNF expansion of )()(),,( zyyxzyxF

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

0 0 0

Example: Find the sum-of-products expansion of the Boolean function that equals 1 if and only if

Laws of Boolean Algebra Concluded

DeMorgan’s Laws

Absorption Laws

Unit Property

Zero Property 0

1

)(

xx

xx

xyxx

xxyx

yxyx

yxxy

Functional Completeness• A set of operators on an algebra is said to be

functionally complete if any function of any degree on that algebra can be expressed in terms of those operators

• The set is functionally complete in any boolean algebra.

• But since , so is• Also, since , is also

functionally complete

},,{

𝑥𝑦=¿𝑥+𝑦=¿

Example: Express the Boolean function using only the operators and .

The NAND Operation

yx |

The NOR Operation

yx

Example:

1. Express the Boolean function using only the operator and using only the operator .

11.3 Logic Gates

• Boolean algebra is the algebra of circuits.• The elementary operations of the algebra

correspond to circuit elements called gates.

Basic Types of Gates

Combinations of gates

• Branching and multiple inputs

xyyx

xyxxy

)(

)(

Example:

Example

• Three switches x, y, and z, controlling a light

x y z F(x,y,z)

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

0 0 0

Half-Adder Circuit

Full Adder

11.4 Minimization of Circuits

• Minimizing a circuit is minimizing the number of gates necessary to achieve the required outputs

• Equivalent to minimizing the Boolean function, i.e. to finding the least number of Boolean operations needed to compute the function

• Recall that any such function can be expressed as a sum of minterms– Note that the number of possible minterms is

exponential in the number n of variables.– Note also that the number of functions is exponential in

the number of minterms.

Karnaugh Maps for Two Variables

• Variables x and y• Label rows with x and x, columns with y

and y

Another Example𝐹 (𝑥 , 𝑦 )=𝑥 𝑦+𝑥 𝑦

Karnaugh Maps (K-Maps) for Three Variables

• Variables x, y, and z• Use x and x as row labels• Use all possible products of y and z

literals, arranged in a Gray code, as column labels

• The geometric picture is that of a band, since the last and first cells in each row are to be considered adjacent

Examples

𝐹 (𝑥 , 𝑦 ,𝑧 )=𝑥𝑦 𝑧+𝑥 𝑦 𝑧+𝑥 𝑦𝑧+𝑥 𝑦 𝑧

𝐹 (𝑥 , 𝑦 ,𝑧 )=𝑥𝑦𝑧+𝑥 𝑦 𝑧+𝑥 𝑦𝑧+𝑥 𝑦 𝑧

Implicants, Prime Implicants, and Essential Prime Implicants

Example𝐹 (𝑥 , 𝑦 ,𝑧 )=𝑥𝑦𝑧+𝑥 𝑦 𝑧+𝑥 𝑦𝑧+𝑥 𝑦 𝑧+𝑥 𝑦 𝑧

K-Maps for Four Variables

• Variables w, x, y, and z• Use all possible products of w and x

literals, arranged in a Gray code, as row labels

• Use all possible products of y and z literals, arranged in a Gray code, as row labels

• Geometric picture is that of a torus

Example𝐹 (𝑤 , 𝑥 , 𝑦 , 𝑧 )=𝑤𝑥𝑦𝑧+𝑤𝑥𝑦 𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥𝑦𝑧+𝑤𝑥𝑦𝑧+𝑤𝑥𝑦𝑧

Another Example𝐹 (𝑤 , 𝑥 , 𝑦 , 𝑧 )=𝑤𝑥𝑦 𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥 𝑦 𝑧

𝐹 (𝑤 , 𝑥 , 𝑦 , 𝑧 )=𝑤𝑥𝑦𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥𝑦 𝑧+𝑤𝑥𝑦𝑧+𝑤𝑥𝑦 𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥 𝑦 𝑧+𝑤𝑥𝑦 𝑧

A light controlled by 4 switches:

The Quine-McCluskey Method

• Map each min-term into a bit string. E.g. map wxyz to 0110.

• Generate implicants as bit strings with wild card characters, such as 0–01

• Prime implicants are those which are not generalized by any other implicant

• Essential prime implicants are those which generalize a min-term not generalized by any other implicant

Example

wxyz, wxyz, wxyz, wxyz, wxyz, wxyz, wxyz

wxyz, wxyz, wxyz, wxyz, wxyz, wxyz, wxyz

Check with Karnaugh Map

Linkhttp://www.mathcs.bethel.edu/~gossett/DiscreteMathWithProof/QuineMcCluskey.html