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The Green Bay Packers enjoyed a perfect first quarter of the 2020 season, cruising to a 4-0 start behind an incredible offense that scored 152 points without turning the ball over.
Now, Matt LaFleur’s team is coming off the early bye week and preparing to travel to Tampa Bay, where they’ll face Tom Brady and the Buccaneers to start the second quarter of 2020.
Can the Packers keep rolling? Improvement in some specific areas might be required, especially if the Packers want to survive an upcoming stretch featuring three road games in four weeks.
Here’s a few things LaFleur and the Packers can improve coming out of the bye:
1 Tackling
AP Photo/Butch Dill
Tackling wasn’t just a problem against Alvin Kamara in New Orleans. The Packers have missed 36 tackles in just four games, according to Pro Football Focus. The secondary has been the worst offender, by far. Darnell Savage (7), Kevin King (4), Will Redmond (3), Adrian Amos (3) and Josh Jackson (3) account for 20 of the 36 misses. There have been far too many misses in the passing game. Expect teams to keep challenging the Packers to make tackles in space until they prove they can do it consistently. And tackling better could certainly help the run defense, which is also on this list.
2 Fourth-and-short
Green Bay Packers running back Aaron Jones (33) celebrates scoring a touchdown against the Minnesota Vikings during their football game Sunday, Sept. 13, 2020, at U.S. Bank Stadium in Minneapolis, Minnesota. Green Bay won 43-34.
The Packers have been aggressive on fourth down, going for it on all six opportunities when needing fewer than three yards on fourth down this season. That’s great, but they could get better on 4th-and-1. LaFleur’s team has only converted two of the five opportunities when needing just one yard, including two from the 1-yard line. Aaron Jones and Jamaal Williams have both been stuffed on 4th-and-1, and Davante Adams dropped a touchdown on fourth down in the opener. The misses are at least partially why the Packers rank 15th in the NFL in scoring touchdowns in the red zone.
3 Pass-rushing pressure
AP Photo/Mike Roemer
The Packers were one of the NFL’s best at creating pressure on quarterbacks last season, but the pass-rush has been somewhat quiet to start 2020. Not having Kenny Clark for the part better of three and a half games certainly hasn’t helped, and both Za’Darius Smith and Rashan Gary have battled their own injuries. However, Preston Smith has struggled to be productive, and Za’Darius Smith hasn’t been as dominant (although he came alive against the Falcons in Week 4). Overall, the Packers have just 47 pressures in four games, putting them on pace for 188 in 2020. They had 302 in 16 games last season. Can they turn up the heat on quarterbacks the rest of the way? Pressuring Tom Brady will be necessary Sunday.
4 Takeaways
AP Photo/Morry Gash
The Packers had 25 takeaways last season, including eight in the first four games. This season, Mike Pettine’s defense has only three, and doesn’t have a single game with more than one. Interceptions (two) are way down. The Packers have been great at forcing three-and-outs, and it helps that the offense hasn’t had a giveaway, but turnovers are so often critical plays. In fact, all three of the Packers’ takeaways this season have been huge, game-turning plays. Turnover production can vary season-to-season, but the Packers are built to be an opportunistic and disruptive defense that will likely need more takeaways over the final 12 games.
5 Red zone defense
Jesse Johnson-USA TODAY Sports
The Packers were great in the red zone last season, ranking fourth in the NFL in points allowed per red zone trip at 4.35. This season, the Packers are allowing 5.85 points per red zone trip, which ranks 30th in the NFL. The opponent’s touchdown percentage (69.2) is up, and the Packers haven’t gotten any of the key takeaways inside the 20-yard line that helped power some of the defense’s red-zone efficiency last season. This group has been breaking – and not just bending – too much to start 2020.
![Things 3 3 1 2 Things 3 3 1 2](https://classicreload.com/sites/default/files/dosx-thinking-things-2-screenshot.png)
6 Run defense
AP Photo/Tom Lynn
Teams have only run 88 times against the Packers through four games, helping mask some of the defense’s issues against the run. The Packers are still allowing 4.8 yards per carry (25th in NFL) and rank 26th in DVOA against the run. Red giant shooter suite 13 1 13 download free. At some point, teams are going to take the Packers’ dare and actually commit to the run within the flow of a normal game script. Holding big leads in the second halves of games – eliminating the opponent’s ability to consistently run the ball – have really helped the Packers. That won’t always be the case over the final 12 games.
Forget everything you know about numbers.
In fact, forget you even know what a number is.
This is where mathematics starts.
Instead of math with numbers, we will now think about math with 'things'.
Definition
What is a set? Well, simply put, it's a collection.
First we specify a common property among 'things' (we define this word later) and then we gather up all the 'things' that have this common property.
For example, the items you wear: hat, shirt, jacket, pants, and so on.
I'm sure you could come up with at least a hundred.
This is known as a set.
Or another example is types of fingers. This set includes index, middle, ring, and pinky. |
So it is just things grouped together with a certain property in common.
Notation
There is a fairly simple notation for sets. We simply list each element (or 'member') separated by a comma, and then put some curly brackets around the whole thing:
The curly brackets { } are sometimes called 'set brackets' or 'braces'.
This is the notation for the two previous examples:
{socks, shoes, watches, shirts, ..}
{index, middle, ring, pinky}
{index, middle, ring, pinky}
Notice how the first example has the '..' (three dots together).
The three dots .. are called an ellipsis, and mean 'continue on'.
So that means the first example continues on .. for infinity.
(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)
So:
- The first set {socks, shoes, watches, shirts, ..} we call an infinite set,
- the second set {index, middle, ring, pinky} we call a finite set.
But sometimes the '..' can be used in the middle to save writing long lists:
Example: the set of letters:
{a, b, c, .., x, y, z}
In this case it is a finite set (there are only 26 letters, right?)
Numerical Sets
So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?
Set of even numbers: {.., −4, −2, 0, 2, 4, ..}
Set of odd numbers: {.., −3, −1, 1, 3, ..}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ..}
Positive multiples of 3 that are less than 10: {3, 6, 9}
Set of odd numbers: {.., −3, −1, 1, 3, ..}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ..}
Positive multiples of 3 that are less than 10: {3, 6, 9}
And so on. We can come up with all different types of sets.
We can also define a set by its properties, such as {x|x>0} which means 'the set of all x's, such that x is greater than 0', see Set-Builder Notation to learn more.
And we can have sets of numbers that have no common property, they are just defined that way. For example:
{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
Are all sets that I just randomly banged on my keyboard to produce.
Why are Sets Important?
Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.
Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets.
Universal Set
At the start we used the word 'things' in quotes. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to our question. |
In Number Theory the universal set is all the integers, as Number Theory is simply the study of integers. |
But in Calculus (also known as real analysis), the universal set is almost always the real numbers. |
And in complex analysis, you guessed it, the universal set is the complex numbers. |
Some More Notation
When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set. So for example, A is a set, and a is an element in A. Same with B and b, and C and c. |
Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?
Also, when we say an element a is in a set A, we use the symbol to show it.
And if something is not in a set use .
And if something is not in a set use .
Example: Set A is {1,2,3}. We can see that 1 A, but 5 A
Equality
Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!
Example: Are A and B equal where:
- A is the set whose members are the first four positive whole numbers
- B = {4, 2, 1, 3}
Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!
And the equals sign (=) is used to show equality, so we write:
A = B
Example: Are these sets equal?
- A is {1, 2, 3}
- B is {3, 1, 2}
Yes, they are equal!
They both contain exactly the members 1, 2 and 3.
It doesn't matter where each member appears, so long as it is there.
Subsets
When we define a set, if we take pieces of that set, we can form what is called a subset.
Example: the set {1, 2, 3, 4, 5}
A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.
But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.
In general:
A is a subset of B if and only if every element of A is in B.
So let's use this definition in some examples.
Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?
1 is in A, and 1 is in B as well. So far so good.
3 is in A and 3 is also in B.
4 is in A, and 4 is in B.
That's all the elements of A, and every single one is in B, so we're done.
Yes, A is a subset of B
Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.
Let's try a harder example.
Example: Let A be all multiples of 4 and B be all multiples of 2.
Is A a subset of B? And is B a subset of A?
Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.
The sets are:
- A = {.., −8, −4, 0, 4, 8, ..}
- B = {.., −8, −6, −4, −2, 0, 2, 4, 6, 8, ..}
By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:
So:
A is a subset of B, but B is not a subset of A
Proper Subsets
If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion. Pdf reader pro 2 7 2007.
Let A be a set. Is every element of A in A?
Well, umm, yes of course, right?
So that means that A is a subset of A. It is a subset of itself!
This doesn't seem very proper, does it? If we want our subsets to be proper we introduce (what else but) proper subsets:
A is a proper subset of B if and only if every element of A is also in B, and there exists at least one element in B that is not in A.
This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.
Example:
{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.
Example:
{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.
Notice that when A is a proper subset of B then it is also a subset of B.
Even More Notation
When we say that A is a subset of B, we write A B.
Or we can say that A is not a subset of B by A B ('A is not a subset of B')
When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.
Empty (or Null) Set
Things 3 3 1 2 Pottery Tea Cup
This is probably the weirdest thing about sets.
As an example, think of the set of piano keys on a guitar.
'But wait!' you say, 'There are no piano keys on a guitar!'
And right you are. It is a set with no elements.
This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.
It is represented by
Or by {} (a set with no elements)
Some other examples of the empty set are the set of countries south of the south pole.
So what's so weird about the empty set? Well, that part comes next.
Empty Set and Subsets
So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?
Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A. But what if we have no elements?
It takes an introduction to logic to understand this, but this statement is one that is 'vacuously' or 'trivially' true.
A good way to think about it is: we can't find any elements in the empty set that aren't in A, so it must be that all elements in the empty set are in A.
So the answer to the posed question is a resounding yes.
The empty set is a subset of every set, including the empty set itself.
Order
No, not the order of the elements. In sets it does not matter what order the elements are in.
![Things Things](https://files.liveworksheets.com/def_files/2020/5/28/528020507285983/528020507285983001.jpg)
When we say order in sets we mean the size of the set.
Another (better) name for this is cardinality.
A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).
For finite sets the order (or cardinality) is the number of elements.
For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.
Things 3 3 1 2 A Rational Number
Arg! Not more notation!
Things 3 3 1 2 45
Nah, just kidding. No more notation.
by Ricky Shadrach and |