Copyright 2021 Carl Dennis Pineda
About Me
Hi there! I am Carl Dennis Pineda, a student from Rizal Technological University who is currently a freshman in the course Information Technology in the section CEIT-37-201A.
Carl Dennis Pineda
Carl is a huge person, yes double dimensions huge. He actually looks scary, looks like it but he's harmless, and its a guarantee. He is the president of CEIT - 37 - 201A, and a simple student in his course. He studied Senior High School in the Rizal Technological University as well but in Pasig Campus. He is just a bored guy who used to watch 1 season of anime per day but now can't even finish 1 episode. He is a grammar nazi, but he seems so gentle even at his size mostly when you knew him, can't even kill a little fly, a real pacifist. An average intellect mind that is properly functioning depending on his mood. Can do some bits of programming, coding, designing(well this is technically his kryptonite), and a bit of logistics with mathematics. His favorite subjects before were computer-related subjects, math, and physics. He aspires to become a fullstack developer, if not he hopes that he'll just become a mafia boss.
About the Page
This site was done for the final requirement of Discrete Math from Professor Jenelyn Luna, and it includes different sections. The collection of the solution for the Activities, Lessons that were included in the Video Lesson, and the things that was learned during the semester in the subject. This is for compliance for the final requirement E-Portfolio.
What have I learned in Discrete Math?
Along the way, the subject has taught me different aspects and it is not limited to the lessons itself. It also taught something that can be applied not only in the lesson but as well in life.Fundamentals
I learned that learning fundamentals is so important so that anywhere or any problem you might face even at such complexity, you will still be able to do it without much issues when you know the fundamentals because you start lower.
Organization
I learned that with proper organization, not only of objects but also thoughts, you can solve any problem at any rate when you have your thoughts organized because when its more compact, you think more logically. The firmer the thoughts, the better.z
Hardwork
I learned that with hardwork, you can make the dream work. As long as you strive for something, a problem hinders you and can't solve it, try different ways to find answers, understand it and you'll less likely find yourself in the same problem again. The rewards are just expanding as you work harder.
Basics
If the ways you used to have isn't working properly, then start climbing back to basics, before ever facing complex things. If you haven't mastered the basics, you don't know the fundamentals, well you will find it hard at complex problems. When everything else fails, standardize and look at the basics.
Activities
This section shows the solutions on the activities we have done. Click the search icon to see the pictures in a better view.
Quiz #1
Drawing Logic DiagramThis were one of the first Logic Diagrams I was making and was actually enjoying drawing them on paint app since it was fun after all.
Quiz #1
Drawing Logic DiagramAnother one from the first batch I was making and was actually enjoying myself on. Drawing Logic Diagrams were fun for me.
Quiz #1
Truth TableTruth table was good for me, however sometimes it can be tricky since I have a really low attention span. So if I have lost my focus, I have to start from the top again
Quiz #1
Drawing Logic DiagramThis is one of the first but is more complicated than the first ones I have drawn, yet I still have enjoyed this one for sure, was just a little pain since it was tricky but liked it too
Quiz #1
Combinatory CircuitThis was a wild one, making combinatory circuits isnt too easy, it is a little bit complicated, but still good and was challenging for me
Task Perfromance #1
Logical EquivalenceLogical Equivalence was a bit tricky to me, since it involves tables and that is the hardest part for me
Task Performance #1
Logical EquivalenceAgain, another logical equivalence which is not much as well since it has a little table
Task Performance #1
Logical EquivalenceSolving logical equivalence with these types of solutions make me enjoy it for real
Task Performance #1
Logical EquivalenceThis solution was a really long one but I really enjoyed it since it has some algebraic feeling.
Task Performance #2
Boolean LawBoolean Law was actually a good ride for me, it makes me remember the pain from thinking if your expression is already simplified or not
Task Performance #2
Boolean LawThe boolean laws are getting more complicated and i really enjoy it since it gives me algebraic feeling, since ofcourse it is boolean algebra.
Task Performance #2
Drawing Logic DiagramI really like the inverters here, and really just enjoying myself drawing this using the paint app in windows.
Task Performance #2
Combinational CircuitMaking Combinational Circuits can be tricky but sure is fun.
Task Performance #2
Logical EquivalenceTaking these Logical Equivalence questions are fun for me, but again, always tricky, which is im working on.
Assignment #3
Truth TableWhile there were more logical equivalence I do, the more I get used to it and it gets less tricky.
Assignment #5
Boolean LawThe more boolean algebra I do, the more I am enjoying it and I am getting used of it.
Assignment #5
Boolean LawAt first, I find using distributive law a very tricky one, but since I had an encounter for a while already and getting used of it
Assignment #5
Boolean LawI remember this item because I thought I already got the whole concept about the absorptive law but after this I actually did
Assigmnment #5
Boolean LawI am actually enjoying these laws, however I hope im not wrong because that'll take toll on my confidence about this
Assignment #6
Drawing Logic DiagramI am really enjoing myself with all of these diagrams, and have finally memorized the diagrams to draw
Assignment #6
Drawing Logic DiagramLater on, I am realizing, my illustrations look kinda ugly sometimes, but I'm working on it somehow
Inverse Function
Have you ever encountered an error? Have you ever done something and reversed it? Therefore you already applied Inverse Function. In this section, inverse function will be given discussion at, as part of the final requirement for Discrete Math.What is inverse function?
Inverse function is a function where you will invert or reverse a function
As you can see, when a function was done, as per in the picture, the apple was transformed into a banana, and as inverse the banana was transformed back into the apple.
Basically, inverse function is inversing your function Inverse function is a function where there is a reversal of a function. It can also be defined as undo of the action of the another function. It is denoted as f(x)-1
Properties of Inverse Function
•f and g are both one to one functions. When we say f and g, these are the two functions that we have i.e. f(x) and g(y). One to one functions map each value in their domain to exactly one value in the co-domain(range)
Image: One to one function map
•Co-domain of f is the domain of g and vice versa
Solving of Inverse Function
Problem 1 : x + a = y
x + a = y
Now we have to equate this into x to get the INVERSE of the function. Therefore we can transpose a to the side of y. Now we have :
x = a - y
As simple as that, works like a charm doesnt it? Now lets head to the next example :
Problem 2 : mx = y
mx = y
Always check the operation, we know that the reverse of addition is subtraction and vice versa; the reverse of multiplication is addition and vice versa
Now, we can divide both sides by m, so that it will be equated to x as the inverse of y.
x = y/m
And that is the inverse of the function!
Now let us try a much more complicated example
Problem 3 : a(x) = 7x + 3
a(x) = 7x + 3
Look complicated, isnt it? But let us take a look on how to invert it so that we can take the inverse of the given function
First, we have to rewrite the problem into linear equation and switch x and y
x = 7y + 3
Now after rewriting the expression, we can transpose 3 to the side of x.
x-3 = 7y
After which, we can now divide both sides to 7 to equate it to y
(x-3)/7 = y
Therefore, we can write it back to function but since its inverse, it is denoted by F(x)-1:
a(x)-1 = (y-3)/7
Remember : Inverse Functions are denoted as f(x)-1.
And now, we can check if it is the inverse of the function
Assuming that x = 1
a(1) = 7x + 3
We can substitute the function of x to the x in the expression which is 7x + 3
a(1) = 7(1) + 3
a(1) = 7 + 3
a(1) = 10
Now, we can see that the answer is 10, then we can check the other one
a(x)-1 = (y-3)/7
Now we have to put the answer in the function to the inverse function which in this case, is 10 which now will be :
x = (10-3)/7
x = 7/7
x = 1
Now as we can see, it came back to the function 1 that we used. It means its the inverse of the function.
OR
We can also use the same variable which in our example is 1, so now it can be :
a(x)-1 = (y-3)/7
Rewrite the function into y since it is an inverse
x = (y-3)/7
1 = (y-3)/7
Now we can multiply both sides by 7 so we can cancel out the denominator.
7 = y-3
Then transpose -3 to the side of 7 to equate it into y
7 + 3 = y
10 = y
y = 10
As you can see, 10 was the result of the first function, and now we also had it in this proofing, which means it is the inverse of the function.
That is it for the lesson of Inverse Function!