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LU / Electrical Engineering / ENGE 201 / List the rules of simplification.

# List the rules of simplification. Description

##### Description: These notes cover everything we learned in Week 4.
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ENGE 201

• Theories of Boolean algebra are used to minimize a boolean expression to reduce the number of logic gates in a network
• Make sure you can make a boolean expression from a truth table

• Theorems!

X.x = 0                x.y - x + y                all are on page 99Don't forget about the age old question of The most common arrangement of protein coding genes is?

X + x = 1        x + y = x.y

X (+) x = 1        x + xy = xWe also discuss several other topics like What is the 2nd order kinetics?
If you want to learn more check out Explain what the chain rule equation is.
Don't forget about the age old question of Why is adjusting entries important?

(x + y)(x + z) = x + yz

X.x + x.y + x.7 + y.z

X + xy + xz + yz

X + xz + yzWe also discuss several other topics like What is the capital of Australia?

X + yz

X + xy = x + y and x+ xy = x + y and x + xy = x + yWe also discuss several other topics like What refers to computer algorithms that analyze info to discover previously unknown and potentially useful information including relationships and patterns?

Simplification of combinational logic circuits using Boolean algebra

• 1rst step = write the equation
• Apply laws to decrease number of terms and variables
• Remove parenthesis / brackets

Example:

AB + BC (B + C)

AB + B.B.C + B.C.C                distribute

AB + BC + BC                AA = A

AB + BS                        A + A = A                SOP form

B (A + C)                        factor                        POS form

Rules of Simplification

1. Groups must be made from same variable

Example: (1 and 1), not (1 and 0)

1. Groups may be horizontal or vertical but not diagonal
2. The number of cells in a group must be a power of 2
3. Each group should be as large as possible resulting in as few groups as possible
4. Each will containing a 1 must be part of a group if possible
5. Groups may overlap

Multiple groups

• Each groups must have at least one cell that does not belong to any other group
1. Groups may wrap around the table

Example: this is one groups

Example: all 4 corners count

Complete Simplification Process

• When a variable appears in both complemented and truth form, it is eliminated

Example:

x = ABCD + ACD + BD

**For every group, there’s a term**

ENGE 201

y

minterm

maxterm

ABCD + ABCD + ABCD + ABCD + ABCD

(A + B C D)(A + B + C +D)(A + B + C + D)

ABCD + ABCD + ABCD + ABCD +ABCD

ABCD + AB(CD + CD) + AB (CD + CD)

ABCD + (AB + AB)(CD + CD)

**Practice Simplifying**

Karnaugh maps

• Can be used for many variables but typically limited to 5 or 6

Construction

• Each cell represents a line in the truth table (in TITLE)
• The coordinates of any 2 adjacent cells differ by only one variable, that why you do back - skip

**Look at chart on page 109

• When a letter is barred, that’s a 0 Example: A,B
• When a letter is not barred, that’s a 1 Example: A,B

Example:

(A + B)(A + C)

AA + AB + AC + BC                        AA = A

A + AB + AC + BC                        A + AB = A

A + AC + BC                                A + AC = A

A + BC

Example:

ABC + ABC + ABC + ABC

BC (A + A) + ABC + ABC                A + A = 1

BC + ABC + ABC                        1A = A

B (C + AC) + ABC                        factor

B (C + A) + ABC                        A + AB = A + B rule

BC + BA + ABC                        distribute

B (C + A) + AVC                        factor

BC + A (B + C)                        A + AB = A + B rule

3C + AB + AC                        distribute

Example:

A + BC + AB                                breaking longest bar

(A + BC) (AB)                                A = A

(A + BC) (AB)                                distribute

AAB + BCAB                                AA = A, AA = 0

AB + 0

AB

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