Package 10
1. Logic Networks
A logic network is a representation of systems using switches, valves, taps, and gates.
Example: A water-pipe system with taps illustrated with diamond shapes.
Water flows only when the correct combination of taps is open:
Water flows when either tap A or B is open, and tap C must be open.
2. Logic Operators and Functions
The four primary logical operators are:
AND: Z = A x B
Example: Two taps in series must both be open for water (Z) to flow.
OR: Z = A + B
Example: Two taps in parallel; either A or B must be open for water (Z) to flow.
XOR (exclusive OR): Z = A Ф B
Example: Water flows if either A or B is open, but not both.
NOT: Z = ¬A
Example: If tap A is closed and B is open, water (Z) will flow.
A logic function mathematically represents a system, e.g., Z = (A + B) x C.
3. Truth-Tables & Solving Logic Problems
Truth-Tables are used to test logic functions; they outline the possible states of inputs and outputs.
With three taps A, B, and C, each can be either CLOSED (0) or OPEN (1), leading to 2^3 = 8 possible combinations:
A B C | Z
0 0 0 - 0
0 0 1 - 0
0 1 0 - 0
0 1 1 - 1
1 0 0 - 0
1 0 1 - 1
1 1 0 - 0
1 1 1 - 1
Logic Problems can be solved using truth-tables by analyzing portions of complex functions.
Example Logic Function: Z = A x C + B x C
Recreate truth-table for this function to derive outputs based on valid combinations of A, B, and C.
4. Boolean Algebra
Follows the same rules as mathematical algebra:
First resolve brackets, then multiplication, then addition.
Example Logic Functions:
Z1 = (A + B) x C
Z2 = A x C + B x C
Both expressions yield the same results (Z), demonstrating equivalence in Boolean algebra.
5. Binary Number System
Review of the decimal number system with ten states (0-9).
Example: 1978 = 1x10^3 + 9x10^2 + 7x10^1 + 8x10^0.
Introduction to the binary number system, which has two states (0-1):
Example: Binary 10111 = 1x2^4 + 0x2^3 + 1x2^2 + 1x2^1 + 1x2^0 = 23.
Practical representation of binary: OFF/ON or CLOSED/OPEN states.
Truth-tables for previous slides correspond to decimal numbers from 0 to 7, illustrating binary representations.