LAB 10 - LOGIC GATES AND BOOLEAN

Purpose of AND gate

Return 1 if both inputs are 1

Purpose of OR gate

Return 1 if either input is 1 or both are 1

Purpose of NOT gate

Inverse

Purpose of NOR gate

Combination of NOT and OR: return 1 if both inputs are not 1.

Purpose of NAND gate

Combination of NOT and AND: return 1 if either input is 0.

Purpose of XOR gate

• Used to perform modulo sum.

• To check difference:

  • If both inputs are odd → return 1.

  • If both inputs are similar → return 0.

Purpose of XNOR gate

• Used to compare 2 inputs.

• Return 1 if both inputs are equivalent.

• Return 0 if both inputs are different.

AND gate

OR gate

NOT gate

NOR gate

NAND gate

XOR gate

XNOR gate

Combinational system

• Has no memory component.

• Output depends only on the present input and present state.

Sequential system

• Store and use previous state information to determine their next state.

• Produce output based on current input and previous output variables.

• Include memory elements.

The result of X.1 and X + 0

X and X

The result of X.0 and X + 1

0 and 1

The result of X.X’ and X + X’

0 and 1

The result of X.X and X + X

X and X

The result of X + X.Y and X(X + Y)

X and X

3 methods of simplification

Boolean, Karnaugh, and McCluskey

Truth table

• (n + 1) columns with the last column as the ouput.

• 2^n rows of each combination of variables

Cells of a Karnaugh map

• Each cell is a row of combination of variables in truth table.

• Each cell stores the output of f() at the corresponding row in the truth table.

Adjacent cells in Karnaugh map

Must only different in 1 variable when it comes to 2 adjacent cells.

Loop characteristic

Cell at the head and tail of column/row can be adjacent of each others since they are only different in 1 variable.

How many cells can we group in a group in Karnaugh map?

2^i cells, meaning that we can group (2, 4, 8, 16, …) cells.

The process of simplifying in Karnaugh map

1. Group those cells that = ‘1’.

2. Look for variable that change throughout each cell. If it changes, get rid of it.

3. Look for variable that does not change, keep it.

4. Then we can have a simplified function by summing up the simplified terms.