what does set homework mean

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• Thread starter sun 94
• Start date Jun 20, 2010
• Jun 20, 2010

What does it mean by ' to do set work ' in the following sentence? Her portfolio shows that she does set work given by her lectures.  

Senior Member

Is that the actual sentence, or a paraphrase? What's the context? By itself, "set work" could mean work creating sets for theater, movies and/or TV. But the "given by her lectures" part makes no sense to me.  

"Set work" can mean "assigned work" -- is this another way of saying "homework"?  

sun 94 said: What does it mean by ' to do set work ' in the following sentence? Her portfolio shows that she is set to work given by her lectures. Click to expand...

k-in-sc said: Is that the actual sentence, or a paraphrase? What's the context? By itself, "set work" could mean work creating sets for theater, movies and/or TV. But the "given by her lectures" part makes no sense to me. Click to expand...

sun 94 said: What does it mean by ' to do set work ' in the following sentence? Her portfolio shows that she does set work given by her lectures. Click to expand...

JudeMama said: To me this could mean: She does the homework (set work/ the work that was set for her to do) assigned (given) by her lectures (classes/the curriculum at her school). Click to expand...

Have you never seen school classes referred to as lectures? Granted, "given in her lectures" would be better construction.  

Set Symbols

A set is a collection of things, usually numbers. We can list each element (or "member") of a set inside curly brackets like this:

Common Symbols Used in Set Theory

Symbols save time and space when writing. Here are the most common set symbols

In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}

How to set homework: simple tips for teachers

As students grow older, most of them face a similar situation: an increase in the amount of homework that they are expected to do.

Of course, there’s a lot of variation in how kids react to homework, and how parents handle the requirement: some children will enjoy it and do it without being asked, while others need more prodding. Some parents will be real sticklers about when and how homework is done, whereas others may let kids handle it more independently.

Some teachers and parents even reject the whole notion of homework - there is a small but significant movement to “ban homework”  because some feel that the time it consumes and the disagreements it causes at home after school are simply not worth it.

One teacher recently sent a note home to parents saying she was effectively outlawing homework for the year and encouraging families to spend more time together instead.

So, is there any truth, from a scientific perspective, that homework is a waste of time?

Yes and no. Here are four simple recommendations we can make, based on scientific evidence, about homework:

1. The quality of the homework is much more important than the quantity. Look at what your students are doing - and don’t give homework just for the sake of it. If parents perceive the homework you assign to be mere busy work, they may not put much emphasis on its completion at home.

2. The main point of good homework is that it lets children independently practice something they learned at school. As such, the goal should not be necessarily to “get everything right”, but to make an effort to actually attempt the task at hand. Then, children should make sure to get feedback and try to understand where they went wrong.

3. Children should be given roughly 10 mins of homework per night per grade , so a 3rd grade student might spend 30 minutes per night on homework. If you are giving more than this amount, you may want to reconsider your reasons for assigning a heavy homework load.

4. Reach out to parents. Explaining the value of the homework you are setting and encouraging them to have their kids attempt their homework will be a big help to you and your students. Of course, the time it takes different students to complete one piece of work will vary; try to be flexible with respect to the amount of work you expect to be completed. If a child is having a particular difficulty in one area, encourage parents to contact you so it can be addressed in school.

Dr. Yana Weinstein ( @doctorwhy  on Twitter) is an Assistant Professor at the University of Massachusetts Lowell.   She co-runs the  Learning Scientists  blog. Follow the Learning Scientists on Twitter at  @AceThatTest .

For more on the research behind homework, you can read this post  by Dr. Paul Kirschner on their blog.

TES has recently launched the free  TES Teach  app  to help teachers create interactive lessons using digital content.  For more information please visit  www.tes.com/lessons

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1.5: Introduction to Sets and Real Numbers

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Sets - An Introduction

A set is a collection of objects. The objects in a set are called its elements or members . The elements in a set can be any types of objects, including sets! The members of a set do not even have to be of the same type. For example, although it may not have any meaningful application, a set can consist of numbers and names.

We usually use capital letters such as \(A\), \(B\), \(C\), \(S\), and \(T\) to represent sets, and denote their generic elements by their corresponding lowercase letters \(a\), \(b\), \(c\), \(s\), and \(t\), respectively. To indicate that \(b\) is an element of the set \(B\), we adopt the notation \(b\in B\), which means “ \(b\) belongs to \(B\)” or “ \(b\) is an element of \(B\).”  Consequently, saying \(x\in\mathbb {R}\) is another way of saying \(x\) is a real number.

Definition: Subset

Set A is a  subset of Set B if and only if every element in Set A is also in Set B.

In symbols:

\[A \subset B \iff x\in A \rightarrow x\in B\]

Real Numbers and some Subsets of Real Numbers

We designate these notations for some special sets of numbers: \[\begin{aligned} \mathbb{N} &=& \mbox{the set of natural numbers}, \\ \mathbb{Z} &=& \mbox{the set of integers}, \\ \mathbb{Q} &=& \mbox{the set of rational numbers},\\ \mathbb{R} &=& \mbox{the set of real numbers}. \end{aligned}\] All these are infinite sets, because they all contain infinitely many elements. In contrast, finite sets contain finitely many elements. 

We list the natural numbers and integers while defining the rational, real and irrational numbers.

\(\mathbb{N} = \{1, 2, 3,\ldots \}\)

\(\mathbb{z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3,\ldots \}\), definition - rational numbers.

A rational number  is a number that can be expressed as a ratio of two integers (with the second integer not equal to zero). Hence, a rational number can be written as \(\frac{m}{n}\) for some integers \(m\) and \(n\), where \(n\neq0\).

Definition - Real Numbers

The real numbers are the numbers corresponding to all the points on the number line.

Definition - Irrational Numbers

An  irrational number  is a real number that can not be expressed as a ratio of two integers; i.e., is not rational.

Given a set S with a binary operation *, S is closed under the operation * if and only if \(x*y \in S \mbox{ for every }x \in S\mbox{ and for every } y\in S\).

Example \(\PageIndex{1}\)

Suppose you add any two integers together. Will the sum always be an integer?

Yes; that's why the set of integers is closed under addition.

We will use the property that the set of integers is closed under addition, subtraction and multiplication.

Alternate syntax is "closure of integers under multiplication".

This assumption can be used as a reason in an explanation or a proof.

Example \(\PageIndex{2}\)

If \(a,b \in \mathbb{Z}\text{, then }a+b \in \mathbb{Z}\) because ?

The set of integers is closed under addition.

Set Notation

Roster notation.

We can use the roster notation  to describe a set if it has only a small number of elements. We list all its elements explicitly, as in \[A = \mbox{the set of natural numbers not exceeding 7} = \{1,2,3,4,5,6,7\}.\] For sets with more elements, show the first few entries to display a pattern, and use an ellipsis to indicate “and so on.” For example, \[\{1,2,3,\ldots,20\}\] represents the set of the first 20 positive integers. The repeating pattern can be extended indefinitely, as in \[\begin{aligned} \mathbb{N} &=& \{1,2,3,\ldots\} \\ \mathbb{Z} &=& \{\ldots,-2,-1,0,1,2,\ldots\} \end{aligned}\]

In regards to parity , an integer is either even or odd. For now, we will use our common understanding of even and odd and define these terms later in this text. The set of even integers can be described as \(\{\ldots,-4,-2,0,2,4,\ldots\}\).

Set-Builder Notation

We can use a set-builder notation to describe a set. For example, the set of natural numbers is defined as \[\mathbb{N} = \{x\in\mathbb{Z} \mid x>0 \}.\] Here, the vertical bar \(\mid\) is read as “such that” or “for which.” Hence, the right-hand side of the equation is pronounced as “the set of \(x\) belonging to the set of integers such that \(x>0\),” or simply “the set of integers \(x\) such that \(x>0\).” In general, this descriptive method appears in the format \[\{\,\mbox{membership}\;\mid\;\mbox{properties}\,\}.\] The notation \(\mid\) means “such that” or “for which” only when it is used in the set notation. It may mean something else in a different context. Therefore, do not write “let \(x\) be a real number \(\mid\) \(x^2>3\)” if you want to say “ let \(x\) be a real number such that \(x^2>3\).” It is considered improper to use a mathematical notation as an abbreviation.

Example \(\PageIndex{3}\)

Write these two sets \[\{x\in\mathbb{Z} \mid x^2 \leq 1\} \quad\mbox{and}\quad \{x\in\mathbb{N} \mid x^2 \leq 1\}\] by listing their elements explicitly.

The first set has three elements, and equals \(\{-1,0,1\}\). The second set is a singleton set; it is equal to \(\{1\}\).

hands-on exercise \(\PageIndex{1}\label{he:setintro-01}\)

Use the roster method to describe the sets \(\{x\in\mathbb{Z} \mid x^2\leq20\}\) and \(\{x\in\mathbb{N} \mid x^2\leq20\}\).

hands-on exercise \(\PageIndex{2}\label{he:setintro-02}\)

Use the roster method to describe the set \[\{x\in\mathbb{N} \mid x\leq20 \mbox{ and $x=n^2$ for some integer $n$}\}.\]

There is a slightly different format for the set-builder notation. Before the vertical bar, we describe the form the elements assume, and after the vertical bar, we indicate from where we are going to pick these elements: \[\{\,\mbox{pattern}\;\mid\;\mbox{membership}\,\}.\] Here the vertical bar \(\mid\) means “where.” For example, \[\{ x^2 \mid x\in\mathbb{Z} \}\] is the set of \(x^2\) where \(x\in\mathbb{Z}\). It represents the set of squares: \(\{0,1,4,9,16,25,\ldots\}\).

Example \(\PageIndex{4}\)

The set \[\{ 2n \mid n\in\mathbb{Z} \}\] describes the set of even numbers. We can also write the set as \(2\mathbb{Z}\).

hands-on exercise \(\PageIndex{3}\label{he:setintro-03}\)

Describe the set \(\{2n+1 \mid n\in\mathbb{Z}\}\) with the roster method.

hands-on exercise \(\PageIndex{4}\label{he:setintro-04}\)

Use the roster method to describe the set \(\{3n \mid n\in\mathbb{Z}\}\).

Interval Notation

An interval is a set of real numbers, all of which lie between two real numbers. Should the endpoints be included or excluded depends on whether the interval is open , closed , or half-open . We adopt the following interval notation to describe them: \[\displaylines{ (a,b) = \{x\in\mathbb{R} \mid a < x < b \}, \cr [a,b] = \{x\in\mathbb{R} \mid a\leq x\leq b \}, \cr [a,b) = \{x\in\mathbb{R} \mid a\leq x < b \}, \cr (a,b] = \{x\in\mathbb{R} \mid a < x\leq b \}. \cr}\] It is understood that \(a\) must be less than  \(b\). Hence, the notation \((5,3)\) does not make much sense. How about \([3,3]\)? This may be used in some texts to mean \(\{3\}\) but we will only use \(a < b\) for intervals and use roster notation for  single number such as \(\{3\}\).

An interval contains not just integers, but all real numbers between the two endpoints.  For instance, \((1,5)\mathbb \neq \{2,3,4\}\) because the interval \((1,5)\) also includes real numbers such at \(1.276\), \(\sqrt{2}\), and \(\pi\).

We can use \(\pm\infty\) in the interval notation: \[\begin{aligned} (a,\infty) &=& \{ x\in\mathbb{R} \mid a<x \}, \\ (-\infty,a) &=& \{ x\in\mathbb{R} \mid x<a \}. \end{aligned}\] However, we cannot write \((a,\infty]\) or \([-\infty,a)\), because \(\pm\infty\) are not numbers. It is nonsense to say \(x\leq\infty\) or \(-\infty\leq x\). For the same reason, we can write \([a,\infty)\) and \((-\infty,a]\), but not \([a,\infty]\) or \([-\infty,a]\).

Example \(\PageIndex{5}\)

Write the intervals \((2,3)\), \([2,3]\), and \((2,3]\) in the descriptive form.

According to the definition of an interval, we find \[\begin{aligned} {(2,3)} &=& \{x\in\mathbb{R} \mid 2<x<3\}, \\ {[2,3]} &=& \{x\in\mathbb{R} \mid 2\leq x\leq 3\}, \\ {(2,3]} &=& \{x\in\mathbb{R} \mid 2 < x\leq 3\}. \end{aligned}\] What would you say about \([2,3)\)?

Example \(\PageIndex{6}\)

Write these sets \[\{x\in\mathbb{R} \mid -2 \leq x < 5\} \quad\mbox{and}\quad \{x\in\mathbb{R} \mid x^2 \leq 1\}\] in the interval form.

The answers are \([-2,5)\) and \([-1,1]\), respectively. The membership of \(x\) affects the answers. If we change the second set to \(\{x\in\mathbb{Z} \mid x^2\leq 1\}\), the answer would have been \(\{-1,0,1\}\). Can you explain why \(\{-1,0,1\} \mathbb \neq [-1,1]\)?

Example \(\PageIndex{7}\)

Be sure you are using the right types of numbers. Compare these two sets \[\begin{aligned} S &=& \{x\in\mathbb{Z} \mid x^2 \leq 5 \}, \\ T &=& \{x\in\mathbb{R} \mid x^2 \leq 5 \}. \end{aligned}\] One consists of integers only, while the other contains real numbers. Thus, \(S=\{-2,-1,0,1,2\}\), and \(T=\big[-\sqrt{5},\sqrt{5}\,\big]\).

If the membership is not specified, such as: \( \{x \; | \; x^2 \leq 5 \} \)   then it is understood that \(\mathbb{R}\) is the default set that \(x\) belongs to.

hands-on exercise \(\PageIndex{5}\label{he:setintro-05}\)

Which of the following sets \[\{x\in\mathbb{Z} \mid 1<x<7\} \quad\mbox{and}\quad \{x \mid 1<x<7\}\] can be represented by the interval notation \((1,7)\)? Explain.

hands-on exercise \(\PageIndex{6}\label{he:setintro-06}\)

Explain why \([2,7\,]\mathbb \neq\{2,3,4,5,6,7\}\).

hands-on exercise \(\PageIndex{7}\label{he:setintro-07}\)

True or false: \((-2,3)=\{-1,0,1,2\}\)? Explain.

Let \(S\) be a set of numbers; we define \[\begin{aligned} S^+ &=& \{ x\in S \mid x>0 \}, \\ S^- &=& \{ x\in S \mid x<0 \}, \\ S^* &=& \{ x\in S \mid x\mathbb \neq 0 \}. \end{aligned}\] In plain English, \(S^+\) is the subset of \(S\) containing only those elements that are positive, \(S^-\) contains only the negative elements of \(S\), and \(S^*\) contains only the nonzero elements of \(S\).

Example \(\PageIndex{8}\)

It should be obvious that \(\mathbb{N}=\mathbb{Z}^+\).

hands-on exercise \(\PageIndex{8}\label{he:setintro-08}\)

What is the notation for the set of negative integers?

Some mathematicians also adopt these notations: \[\begin{aligned} bS &=& \{ bx \mid x\in S \}, \\ a+bS &=& \{ a+bx \mid x\in S \}. \end{aligned}\] Accordingly, we can write the set of even integers as \(2\mathbb{Z}\), and the set of odd integers can be represented by \(1+2\mathbb{Z}\).

Example \(\PageIndex{9}\)

\[5\mathbb{Z}=\{\ldots , -15, -10, -5, 0, 5, 10, 15, \ldots\}\]

There are three kinds of real numbers: positive, negative and  zero.

Trichotomy Property

For any two real numbers, \(a\) and \(b\) one and only one of these relations is true:

• \(a <b\)
• \(a>b.\)

Exercise \(\PageIndex{1}\)

Determine whether these statements are true or false:

• \(0\in\mathbb{Q}\)
• \(0\in\mathbb{Z}\)
• \(-4\in\mathbb{Z}\)
• \(-4\in\mathbb{N}\)
• \(2\in3\mathbb{Z}\)
• \(-18\in3\mathbb{Z}\)

(a) true (b) true (c) true (d) false (e) false (f) true

Exercise \(\PageIndex{2}\)

• \(\sqrt{2}\in\mathbb{Z}\)
• \(-1\notin\mathbb{Z}^+\)
• \(0\in\mathbb{N}\)
• \(\pi\in\mathbb{R}\)
• \(\frac{4}{2}\in\mathbb{Q}\)
• \(1.5\in\mathbb{Q}\)

Exercise \(\PageIndex{3}\label{ex:prop-03}\)

Explain why \(7\mathbb{Q}=\mathbb{Q}\). Is it still true that \(0\mathbb{Q} = \mathbb{Q}\)?

By definition, a rational number can be written as a ratio of two integers. After multiplying the numerator by 7, we still have a ratio of two integers. Conversely, given any rational number \(x\), we can multiply the denominator by 7, we obtain another rational number \(y\) such that \(7y=x\). Hence, the two sets \(7\mathbb{Q}\) and \(\mathbb{Q}\) contain the same collection of rational numbers. In contrast, \(0\mathbb{Q}\) contains only one number, namely, 0. Therefore, \(0\mathbb{Q}\neq\mathbb{Q}\).

Exercise \(\PageIndex{4}\label{ex:prop-4}\)

Find the number(s) \(k\) such that \(k\mathbb{Z}=\mathbb{Z}\).

Exercise \(\PageIndex{5}\)

(See section on Closure.)

• The set of natural numbers is closed under subtraction.
• The set of integers is closed under subtraction.
• The set of integers is closed under division.
• The set of rational numbers is closed under subtraction.
• The set of rational numbers is closed under division.
• \(\mathbb{Q^*}\) is closed under division.

(a) false (b) true (c) false (d) true (e) false (f) true

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Meaning of homework in English

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• The kids are busy with their homework.
• My science teacher always sets a lot of homework.
• "Have you got any homework tonight ?" "No."
• I got A minus for my English homework.
• For homework I want you to write an essay on endangered species .
• academic year
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homework | American Dictionary

Homework | business english, examples of homework, translations of homework.

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a large flat sea creature that lives in a shell, some types of which can be eaten either cooked or uncooked, and other types of which produce pearls (= small round white precious stones)

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Set Notation

You never know when set notation is going to pop up. Usually, you'll see it when you learn about solving inequalities , because for some reason saying " x < 3 " isn't good enough, so instead they'll want you to phrase the answer as "the solution set is {  x | x is a real number and x < 3 } ". How this adds anything to the student's understanding, I don't know. But I digress....

A set, informally, is a collection of things. The "things" in the set are called the "elements", and are listed inside curly braces.

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For instance, if I were to list the elements of "the set of things on my kid's bed when I wrote this lesson", the set would look like this:

{ pillow, rumpled bedspread, a stuffed animal, one very fat cat who's taking a nap }

Sets are usually named using capital letters. This isn't a rule as far as I know, but it does seem to be traditional. So let's name this set as " A ". Then we have:

A = { pillow, rumpled bedspread, a stuffed animal, one very fat cat who's taking a nap }

The cat's name was "Junior", so this set could also be written as:

A = { pillow, rumpled bedspread, a stuffed animal, Junior }

Sets are "unordered", which means that the things in the set do not have to be listed in any particular order. The set above could just as easily be written as:

A = { Junior, pillow, rumpled bedspread, a stuffed animal }

We use a special character to say that something is an element of a set. It looks like an odd curvy capital E. For instance, to say that "pillow is an element of the set A ", we would write the following:

This is pronounced as " pillow is an element of A ".

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The elements of a set can be listed out according to a rule, such as:

{  x is something on my kid's bed }

A mathematical example of a set whose elements are named according to a rule might be:

{ x is a natural number, x < 10}

If you're going to be technical, you can use full "set-builder notation" to express the above mathematical set. In set-builder notation, the previous set looks like this:

The above is pronounced as "the set of all x , such that x is an element of the natural numbers and x is less than 10 ". The vertical bar is usually pronounced as "such that", and it comes between the name of the variable you're using to stand for the elements and the rule that tells you what those elements actually are.

This same set, since the elements are few, can also be given by a listing of the elements, like this:

{ 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Listing the elements explicitly like this, instead of using a rule, is often called "using the roster method".

Your text may or may not get technical regarding the names of the types of numbers. If it does, these are the symbols to use:

Yes, the symbols require those double-barred strokes for all the vertical portions of the characters.

Sets can be related to each other. If one set is "inside" another set, it is called a "subset". Suppose A = { 1, 2, 3 } and B = { 1, 2, 3, 4, 5, 6 } . Then A is a subset of B , since everything in A is also in B . This relationship is written as:

That sideways-U thing is the subset symbol, and is pronounced "is a subset of".

To show something is not a subset, you draw a slash through the subset symbol, so the following:

...is pronounced as " B is not a subset of A ".

If two sets are being combined, this is called the "union" of the sets, and is indicated by a large U-type character. If, instead of taking everything from the two sets, you're only taking what is common to the two, this is called the "intersection" of the sets, and is indicated with an upside-down U-type character. So if C = { 1, 2, 3, 4, 5, 6 } and D = { 4, 5, 6, 7, 8, 9 } , then:

These are pronounced as " C union D equals..." and " C intersect D equals...", respectively.

Give a solution using the roster method: A = { 1, 2, 3, 4, 5, 6, 7 }, B is a subset of A , the elements of B are even.

The set B is a subset of A , so it contains only things that are in A . The elements of B are even, so I need to pick out the elements of A which are even; these will be the elements of the subset B .

The numbers in A that are even are 2, 4, and 6 , so:

B = { 2, 4, 6 }.

What is the intersection of A = {  x is odd } and B = {  x is between −4 and 6 } , where the elements of the two sets are integers?

Since "intersection" means "only things that are in both sets", the intersection will be all the numbers which are in each of the sets. The elements of B can be listed, being not too many integers:

B = { −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6 }

The elements of A are all the odd integers. There are infinitely-many of them, so I won't bother with a listing. The intersection will be the set of integers which are both odd and also between −4 and 6 . In other words:

{ −3, −1, 1, 3, 5 }

What is the union of A = {  x is a natural number between 4 and 8 inclusive } and B = {  x is a single-digit negative integer } ?

Since "union" means "anything that is in either set", the union will be everything from A plus everything in B . Since A = { 4, 5, 6, 7, 8 } (because "inclusive" means "including the endpoints") and B = { −9, −8, −7, −6, −5, −4, −3, −2, −1 } , then their union is:

{ −9, −8, −7, −6, −5, −4, −3, −2, −1, 4, 5, 6, 7, 8 }

Give a solution using a rule: The set of all the odd integers.

An odd integer is one more than an even integer, and every even integer is a multiple of 2 . The formal way of writing "is a multiple of 2 " is to say that something is equal to two times some other integer; in other words, " x = 2 m ", where " m " is some integer. Then an odd integer, being one more than a multiple of 2 , is x = 2 m + 1 .

So, in full formality, the set would be written as:

The solution to the example above is pronounced as "all integers x such that x is equal to 2 times m plus 1 , where m is an integer".

It's a lot easier to describe the last set above using the roster method:

{ ..., −3, −1, 1, 3, 5, 7, ... }

The ellipsis (that is, the three periods in a row) means "and so forth", and indicates that the pattern continues indefinitely in the given direction. Or, if the dots are between elements, like this:

{ 0, 3, 6, 9, ..., 993, 996, 999 }

...it means that the pattern continues in the same manner through the unwritten middle.

There's plenty more you can do with set notation, but the above is usually enough to get by in most algebra-class circumstances. If you need more, try doing a web search for "set notation".

URL: https://www.purplemath.com/modules/setnotn.htm

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