Is this the correct truth table for product of sums?

[heady lily]

I can see how you would get C=xy but I am not seeing how you can get S=(x+y)(x'+y') since I got zero for all S values.

[gloomy trench]

That's an xor gate, it outputs 1 when exactly one of the inputs is a 1.

[heady lily]

Can I think about the x and y as being the first sum times the second sum?

I am trying thinking about it multiple ways instead of memorizing XOR gates.

[rocky ibex]

Hi

[rocky ibex]

You filled in the truth table incorrectly

[heady lily]

I had a feeling.

[rocky ibex]

Think of XOR as odd parity
It’s 1 when there are odd number of inputs that are 1

In this case 1

[heady lily]

Should I break my truth table down even more?

[rocky ibex]

The S column needs to be changed

0 1 1 0

[heady lily]

Oh, just like the sum of products truth table...?

But I can imagine the sum of products equations way easier.

[rocky ibex]

When you apply x=0 and y = 0
S = 0

x=0,y=1, S=1
x=1, y=0, S=1
both 1, s=0

[heady lily]

So I grab the S values that are 0 and 0 instead of 1 and 1 for the equation.

[rocky ibex]

there’s SOP and POS

if SOP is easier

That’s fine

Just double bar SOP to get POS

You’re double barring this

[heady lily]

Im trying to understand it multiple ways. I think to get the X and Ys where S=0 to get the (x+y)(x'+y') but it doesnt match when I try to get the carry equation. Why does it follow the same formula as SOP?

[heady lily]

I feel like I am just memorizing a formula when I look at this.

[rocky ibex]

It’s better to have 1 method you’re good at and know how to convert to the other forms

So what you want to know is POS

that means when it’s 0 it’s not barred
so you have two combinations for S=0

X=0,Y=0
X=1,Y=1

That correlates to the POS expression of
(X+ Y)(X’ + Y’)

[heady lily]

This doesn't affect the Carry function, right?

[rocky ibex]

You need a foundation to build on
Memorizing a few formulas is okay

What carry function

[heady lily]

C=xy

[rocky ibex]

No

You’ll get it soon just practice more read the book

[heady lily]

Ok thanks

Once I can start picturing the concepts, it helps me a lot. I am not there yet.

[dense trellis]

Logic design

Im really good in it

U still need help?

@heady lily

[leaden drum]

I could just bluntly say (x+y)(x'+y') is the definition of x XOR y.

The idea of the XOR operation is that it's high when exactly one of the inputs is high. so, draw the table:
x | y | x XOR y
0 | 0 | 0
0 | 1 | 1
1 | 0 | 1
1 | 1 | 0

Try messing with the algebra

[heady lily]

I am able to distribute x(y+z)=xy+xz pretty easily; just not the other way around with binary.

[dense trellis]

Whats the question

I see a half adder

Also

S shouldnnot be 0s

All the way

S is an odd function

Its x XOR y

[heady lily]

Maybe I will start a new question? It just that creating the half adder uses a concept from before that I still dont get.

[dense trellis]

Whats the question i can help

[heady lily]

How do I go from F'=AB+CD+BD' to F=(A'+B')(C'+D')(B'+D)?

[dense trellis]

Invert

[heady lily]

I worked with a proof but they went backwards.

[dense trellis]

Invert everything

• becomes × and A becomes A'

Vise versa is true

[dense trellis]

Just do this

Wait

[heady lily]

I know it is Distributive but is there a law for that.

I wont remember to just invert it.

[dense trellis]

The law is u invert its really simple

[heady lily]

I guess practice more DeMorgans Law questions..

[dense trellis]

What it doesnt need practice just make the opposite of what it is

Ill show u how to prove it step by step i guess

@heady lily

Is it more clear now?

[heady lily]

So the word for priming both sides is involution...

[dense trellis]

What

[heady lily]

Or I guess look at the DeMorgan proof.

There are a couple steps skipped in your proof.

[dense trellis]

What is there to prove

Just switch everything to the opposite of what it is

[leaden drum]

demorgans

[heady lily]

Dang it, my book skips the DeMorgan proof

[dense trellis]

Well

Its insanely important

Nands will be ur go to later on

But like its super easy

Dont stress it

Just flip everything to the opposite of what it is

U can do it in 1 step

[heady lily]

Like they have these proofs but skip DeMorgan 😦

[dense trellis]

These proofs dont memorise just understand

Unless ur Dr. Specifically said

Memorise

[heady lily]

I am trying to find DeMorgans proof.

[dense trellis]

Why

[leaden drum]

let's prove demorgans:

Assume(1) (xy)'
assume(2) (x' + y')'
thus, (x -> y')'
thus, x(y'')
thus, xy, but that contradicts (1), so assumption 2 is wrong.
thus, x' + y'

that shows (xy)' implies x' + y'

[dense trellis]

• X+1 =1
• X+0 = x
• (X)(1) = x
• (X).(0) = 0
• X=1 so x' = 0

Theres a few more but these are the main ones

And for proofs i lowkey just read them and thats about it their pretty useless icl

[leaden drum]

if you really want a quick proof, just do it by truth table. the variables only take on a finite set of values: {true, false}

[dense trellis]

Unless Dr. Said otherwise

[leaden drum]

I don't think proofs are useless. If you can't prove a lot of what you use, you have to take a lot of what you use on faith, and that's not good practice.

[dense trellis]

If ur a mathematician or u want to create new theories

Yes

Proofs ar important

If u wanna pass the course and movenon with ur life

Fully understanding proofs are a waste of time icl

Unless Dr. Said otherwise

[leaden drum]

so you move on to a point where you have to take a lot of what you use on faith?

[dense trellis]

Understanding logically ≠ doing proofs

[leaden drum]

producing a faith from scratch yourself, and merely checking someone else's proof are 2 different things.

if someone did the proof for you, it should be very quick to follow it and see that it is true. Doing if from scratch will take a long time and that's not the goal as an engineer.

mathematicians may want to practice proving things from scratch. engineers don't have to do that, but they should be able to follow the proofs they apply later.

[dense trellis]

Bro as i stated before if u wanna become a mathmatecian or you wanna create new theorems then proofs are super important

BUT if u just want to pass the course then understanding the proof is enough

[leaden drum]

yes, understanding the proof is enough.

[dense trellis]

unless stated otherwise by ur dr.

[dense trellis]

Yep

[leaden drum]

and by understanding I mean checking it as if you were a teacher checking homework.

if an alleged proof has a mistake you should be able to see the mistake.

[heady lily]

The thing is im not taking a course... i want to really understand this to make electronics easier

[dense trellis]

There is no mistake in what hes studying so no point

[dense trellis]

Electronics?

Bro

Electronics is pure trigonometry

[heady lily]

Im reading a digital logic book

[dense trellis]

Its full of phasers op amps

Amps

In general

U dont take logic design

This is for logic design and microprocessing

Building logical circuits

[heady lily]

Using logic gates to build a computer

[dense trellis]

U mean the course or electronics as in

Electrical appliances

[heady lily]

Digital Logic and Computer Design. I want to understand how cpus and gpus work

[dense trellis]

Ah

Yep

Do this

Study binary really well

And do assembly

Thats for cpus

For building study this

The order of how u should study is this

@heady lily

1. proofs

2. practice on binary and conversion of bases

3. binary and other base addition subtraction multiplication division...for both integers and non integers

4)study AND, OR, NOT gates and try to solve or create basic circuits....try to create their equations and do their truth tables

5. get really good with truth table so that u can do an equation using a truthtable of 4 inputs

6. Study XOR, NXOR, NAND, NOR

7. study K-MAPS

8. practice really well so u can do 4 input K-maps

9. START CREATING circuits like halfadders, full adders, BCD adder, EXCESS 3 adder, BCD convertor etc etc

10. study sequential circuits

Notice how half adders are step 8

Thats because u need to master the basics before u can start creating circuits

This wont take you more than 1 month btw depends how much time u give it

@heady lily These are my exams

They should be good practice

To enhance ur basics

[heady lily]

Cool. Thanks!