Minimization of Boolean Functions

What are the three Boolean laws you will probably use to minimize Sum Of Product (SOP) functions?
The three Boolean laws you will probably use to minimize SOP functions are:

  1. Distributive law.
  2. Complement law for OR logic.
  3. Identity law.

What is an example of a Sum Of Products (SOP) function written from a truth table?
An example of an SOP function written from a truth table is:

Row
0 0 0 0 1
1 0 0 1 0
2 0 1 0 1
3 0 1 1 0
4 1 0 0 1
5 1 0 1 1
6 1 1 0 1
7 1 1 1 0

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NAND and NOR Logic

What are the two reasons we use NAND and NOR gates instead of AND and OR gates?
We use NAND and NOR gates instead of AND and OR gates because:

  1. NAND and NOR gates require less transistors - CMOS transistors act as inverters already.
  2. NAND and NOR gates can be used to create all other logic gates - They're known as universal gates.

Why do NAND and NOR gates require less circuitry?
NAND and NOR gates require less circuitry because the CMOS transistors in the gates invert the output by default.

Note

Essentially, the CMOS transistors which are used in logic gates invert whatever their input is by default, since that is how they work. To create an AND gate from a NAND, you would have to add two more CMOS transistors to invert the output again.

What is the circuit diagram of a 2-input NAND gate compared with the circuit diagram of a 2-input AND gate?
The circuit diagram of a 2-input NAND gate compared with the circuit diagram of a 2-input AND gate is:
2-Input NAND Gate And 2-Input AND Gate.png

What is the circuit diagram of a 2-input NOR gate compared with the circuit diagram of a 2-input AND gate?
2-Input NOR Gate And 2-Input OR Gate.png

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Building Logic Circuit From SOP / POS

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Example of converting a logic circuit to only use NAND gates

Converting A Logic Circuit To Use NAND Gates Only.png

How do you convert a Product Of Sum (POS) circuit to use NOR gates?
To convert a POS circuit to use NOR gates:

  1. Insert pairs of inverters (which look like bubbles) between logic gates.
  2. Convert all AND gates with inverters on both of their inputs to NOR gates.
Example of converting a logic circuit to only use NOR gates

Converting A Logic Circuit To Use NOR Gates Only.png

Karnaugh Map

What is an example of a Karnaugh map made from a truth table?
An example of a Karnaugh map made from a truth table is:
0 0 1
0 1 1
1 0 1
1 1 0
\ 0 1
0 1 1
1 1 0

What is the simplified Boolean function of the above Karnaugh map?
The simplified Boolean function of the above Karnaugh map is .

Forming K-Map

What are the steps for using a Karnaugh map?
The steps for using a Karnaugh map are:

  1. Focus on the values which are 1 (from the SOP method).
  2. Form groups of adjacent values which are 1. Overlapping groups and groups that wrap around the edges of the table horizontally and vertically are allowed.
  3. Variables which change along a row or column of the map can be ignored.
  4. Variables which stay the same along a row or column of the map can be added together.

Now, Three-Variable K-Map

How do you make a Karnaugh map for functions with more than two variables?
To make a Karnaugh map for functions with more than two variables, you combine variables on a single axis and increment them like a binary number.

What is an example of a Karnaugh map with three variables?
An example of a Karnaugh map with three variables is:

\ 00 01 11 10
0 0 1 1 1
1 0 1 0 0

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Note

When you write the columns and rows for a Karnaugh map, only a single digit can change from the previous column or row's number. Instead of going from 00 to 01 to 10, you have to go from 00 to 01 to 11 to 10. This is called a Gray Code.

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Rules of Grouping

What are the five rules for forming groups of 1's on K-Maps?
The five rules for forming groups of 1's on K-Maps are:

  1. All 1's have to be included in at least one group.
  2. Groups which overlap are valid as long as they only overlap once.
  3. The number of terms in your Boolean function is equal to the number of groups.
  4. Non-adjacent 1's can't be grouped.
  5. It's okay if a single 1 forms its own group as long as there is no other option.

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