What is Discrete Mathematics ?
Discrete mathematics is mathematics that deals with discrete objects.
Discrete objects are those which are separated from (not connected to/distinct
from) each other. Integers (aka whole numbers), rational numbers (ones that
Contents
Often we want to discuss properties/relations common to all propositions. In such a case rather
than stating them for each individual proposition we use variables representing an arbitrary
proposition and state properties/relations in terms of th
proposition
Contents
Sentences considered in propositional logic are not arbitrary sentences but are the ones that are
either true or false, but not both. This kind of sentences are called propositions.
If a proposition is true, then we say it has a trut
Once we have it in this compact form, it is fairly easy to compute S for different values of A, R
and n, though one still has to compute (1 + R)n + 1 . This simple formula represents infinitely
many cases involving all different values of A, R and n. The
Logic is a language for reasoning. It is a collection of rules we use when
doing logical reasoning. Human reasoning has been observed over centuries
from at least the times of Greeks, and patterns appearing in reasoning have
been extracted, abstracted, an
Introduction
Everyone must have felt at least once in his or her life how wonderful it would be if we could
solve a problem at hand preferably without much difficulty or even with some difficulties.
Unfortunately the problem solving is an art at this poin
One of the most important thing, if not the most important, in the university education is the
cultivation of the ability to extrapolate. Obviously we can not anticipate all the problems you are
going to encounter in the future and prepare you for them. S
Example 5
This is a find type problem and "working backward" technique is used.
Problem: Given a 4 quart pail and a 9 quart pail, obtain 6 quarts of water in the 9 quart pail
using these two pails. You can fill or empty the pails and you can have as much
Example 1
This is an example in which you can find a solution once you analyze and understand the
unknowns and data.
Problem: A survey of TV viewers shows the following results:
To the question "Do you watch comedies ?", 352 answered "Yes".,
To the questi
As an example, let us consider a simple problem of investment. Suppose that we invest $1,000
every year with expected return of 10% a year. How much are we going to have after 3 years, 5
years, or 10 years ? The most naive way to find that out would be th
Let us define the meaning of the five connectives by showing the relationship between the truth
value (i.e. true or false) of composite propositions and those of their component propositions.
They are going to be shown using truth table. In the tables P a
You've already seen glimpses of matrices - determinants (for Cramer's
Rule) and Gaussian elimination. Now, we'll see what else we can do
with them.
A matrix is just a rectangular grid of numbers. Keanu Reeves will tell
you otherwise, but don't believe him
Remember that a trinomial has three terms.
Like this guy:
Right now, we're really interested in a certain type of trinomial.
Remember FOIL?
We can multiply
and get
Hey - that's our trinomial from above.
Basically, we'll be figuring out how to "undo" guys
This is a pretty easy method. But, it only works when you can factor
something. As I told you before, except for in math books, things
usually don't factor.
But, this is a good method to help you understand what's going on. so,
here we go!
We're going to
Let's start with an easy one:
Solve
x = 10
We can just look at it and see that
.
But, what if we didn't see that? What would we do?
Here's the Algebra trick:
We'll add
3 to both sides!
*Remember the see saw?
Whatever we do to one side of the equation,
we
Square Roots:
Here are a few square roots:
The first two popped cleanly (because they had perfect squares inside.)
But, the last guy didn't. He's an irrational number. His decimal part
goes on forever and ever and never repeats.
Just like
!
The reason I u
So far, we've learned how to add and subtract matrices (the sizes had to
be the same) and how to multiply a matrix by a scalar. These were really
easy.
Multiplying two matrices is a bit tricky, but, once you get the hang of it,
it's a snap - you just need
Now, we're going to learn how to factor guys with a number in front of
the
.
I'm not going to lie to you - these can be pretty tricky. Some students
don't like these because there is no way to just memorize how to do
them. Each problem is different and yo
Remember from your arithmetic days (or is that daze?) what this
means?
That's three 2's all multiplied together:
As you know by now, in algebra, we work with that unknown guy. Mr. X!
The thing to remember is that x is a number - we just don't know which
o
Remember that monomials are single term critters:
Dividing by these things pops up fairly often in later math classes.
Let's just do one:
Let's rewrite it like this:
Since there is only one term down here, we can break this thing up.
Here's another one:
T
This is really easy, but there is one big thing to remember:
THE SIZES MUST BE THE SAME!
Check it out:
Let's add these guys:
We just add the entries in each spot.
That's it! So, what do ya think - easiest thing on the whole site?
Here's a subtraction guy
But, can we write 12 as the product of just prime numbers?
Well, 3 is a prime number, but 4 isn't.
Can we break 4 into a product of primes?
So,
When we write a number as the product of primes, it's called a prime
factorization.
Smaller numbers are pretty
The P in PEMDAS stands for "parenthesis!"
Parenthesis in math are used to group important things together, so you
always do them first.
Check it out:
Do inside the parenthesis first!
Here's another one:
TRY IT:
%
We use the symbol
for "percent."
The word "percent" really just means "per hundred."
So, 25% means "25 per hundred" which is
.
If you had a pizza that was cut into 100 pieces, 25% of
the pizza would be 25 pieces!
A percent is just another way of countin
First, I need to remind you about something with decimals:
Remember place values?
So.
At first, we'll do this in two steps:
1
Use the word "percent" to convert the percent to a
fraction.
2
Use decimal place values to convert the fraction to
a decimal.
Let
This is super easy!
To change a percent to a fraction, just use what that word "percent"
means. PER HUNDRED!
Let's just do some:
Convert 47% to a fraction:
Convert 91% to a fraction:
Check it out:
Convert 75% to a fraction:
Convert 22% to a fraction:
OK, I know you're dying of curiosity!
What IS the answer to
7? 9?
The last part of PEMDAS is
This stands for
M = Multiplication
D = Division
A = Addition
S = Subtraction
In the word PEMDAS, MD comes before AS . So, the order of
operations rule is that you
What's the answer to this?
Did you get 7?
Do you see how the answer could be 9?
There are two possible ways to do this problem:
Do the addition first:
Do the division first:
Both answers can't be right or we'd always be arguing about the
answers to math p
Remember that fractions are used to count a part of
something.
of this square is red.
Mixed numbers are used when you need to count
whole things AND parts of things at the same time.
Check this out:
How much of these squares is red?
There are 3 whole squa
Divisibility by 2 Test:
A number is divisible by 2 if it is even.
Here's another way to phrase this one:
A number is divisible by 2 if
it's last digit is 0,
2, 4, 6, or 8
Let's do an easy one from our times tables:
Here's a harder one:
What about this one