Logic Composition

Within these castle walls be forged Mavens of Computer Science ...

— Merlin, The Coder

Prerequisites

• W1014 Logic Gates

Introduction

Logic gates can be composed in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic. Combining multiple gates enables us to build complex functionality from simpler building blocks.

Three-Input And Gate

Three Input AND Gate

Consider this three input AND gate internally constructed of two, two-input AND gates. Logically, this gate has three inputs and one output, regardless of how the gate is constructed internally. The truth table for this composition is:

Inputs			Outputs
$A$	$B$	$C$	$Q=A\land B\land C$
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

Helpful Hint

Take the time to recreate this table (and those that follow) on your own

Implication Gate

IMPLY gate

Consider this IMPLY gate internally constructed of a NOT gate and an OR gate. Logically, this gate has two inputs and one output, regardless of how the gate is constructed internally. The truth table for this composition is:

Inputs		Outputs
$P$	$Q$	$Y=\neg A\lor B$
0	0	1
0	1	1
1	0	0
1	1	1

Implication is formally indicated by a forward arrow, i.e. $p\rightarrow q$, meaning, if p is true, then q is also true. As an example, consider the implication, "if it's raining then I'll carry my umbrella". There are four cases, only one of which yields false:

Raining	Equipped with Umbrella	Evaluation of Statement	Explanation
No	No	true	It's not raining, so it doesn't matter if I'm carrying an umbrella or not.
No	Yes	true	It's not raining, so it doesn't matter if I'm carrying an umbrella or not.
Yes	No	false	It is raining, and I am not carrying an umbrella. This contradicts the implication.
Yes	Yes	true	It is raining, and I am carrying my umbrella.

First Steps

This exercise relies on an external website which will enable you to easily experiment with logic gates and form various compositions with the components. The links on this wiki page will preload the relevant gates currently under discussion. For best results, open the links in a separate window so that you can continue to read the wiki while you experiment.

• The "L" indicates a Low voltage which usually represents zero or false
• The "H" indicates a High voltage which usually represents one or true
• The layout flows from left to right.
• Inputs will be leftmost and output will be rightmost.
• Clicking on an input will toggle the input from Low to High or from High to Low
• When an input is changed, the result will instantly impact the rest of the digram.

Input, Output, and an Inverter

Click on the image ("Wire and Inverter") and open the link in a different window, then return to the wiki:

Wire and Inverter

1. Click on the leftmost (input) H in the top diagram which simply represents a wire.
   1. What happens to the H?
   2. Does the color of the wire change?
   3. How is the output impacted?
2. Click on the leftmost (input) H in the bottom diagram which represents an inverter.
   1. What happens to the H?
   2. Does the color of the wire change?
   3. How is the output impacted?

AND Gate

Click on the image ("AND Gate") and open the link in a different window, then return to the wiki:

AND Gate

1. Click on the top L input.
   1. What happens to the top L?
   2. What happens to the output?
2. Click on the bottom L input.
   1. What happens to the bottom L?
   2. What happens to the output?
3. Click several more times on the inputs until you're able to consistently predict the output.