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Day 4

Created Wednesday 26 February 2020

Now we are doing logic gate stuff.

Boolean algebra is a mathematical system...

AND

Boolean product

OR

Boolean sum

NOT

Reverses the value

Boolean function has at least:

it produces an output that is also a member of that set {0,1}

Truth Tables (let n

1

indicate prime)

F(x, y, z) = xz1 + y

There is priority with our operators

NOT takes priority

Most Boolean identities have an AND (product) form as well as an OR (sum) form. We give our identities using both forms

He's going over different laws. I didn't write them all down so some are missing

Absorption Law
Demograns Law

F(x, y, z) = xy + x1z + yz
With the various identities (as we did in discrete mathematics) we can simplify this expression!
In fact, we get just xy + x1z. As such, the third component y was completely unneeded. If we were building this as an actual circuit gate then we'd be wasting with the original function.

DeMorgans law is badass. It's sometimes more economical to build a circuit using the complement of a function (and complementing its result) than it is to implement the function directly.

DeMorgans can be extended to any number of variables. Replace each variable by its complement and change all ANDs to ORs and all ORs to ANDs.
Thus, we find the following complement:
F(x, y, z) = (xy) + (x1y) + (xz1)

Synominous forms are logically equivalent.

There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums.

xy + xz + yz is sum of products.
(x + y) * (x + z) * (y + z) is product of sums.

How do we create a sum-of-products?

Note, there are special symbols we use for OR, AND, etc. Memorize what they look like!

XOR or Exclusive OR

One or the other but not both
0 0 0
0 1 1
1 0 1
1 1 0

NAND and NOR are two very important gates

Easy to manufacture and whatnot.

NAND (a negated AND function)
0 0 1
0 1 1
1 0 1
1 1 0

NOR (negated OR function)
0 0 1
0 1 0
1 0 0
1 1 0

NAND and NOR are known as universal gates because they are inexpensive to manufacture.

Any boolean function can be constructed using only NAND or only NOR gates.

Two NAND gates get us an AND gate.

NOT takes up a single gate

OR is made up of three gates.

Gates can have multiple inputs and more than one output


He is now showing us logisim.

I had to download a jar. It's not in the deb repo

Using project->analyze logic gives us a nice little truth table of our circuit

For homework

He wants us to export the logisim thing as a .jpg. Then upload that to BlackBoard for homework