1.E: Basic Language of Mathematics (Exercises)

Exercise (PageIndex{1}): Statements

Which of the following are statements?

  1. I am here.
  2. Why am I here?
  3. Life is beautiful.
  4. My car is red and my house is yellow.
  5. An integer is even if and only if it is divisible by 2 with no remainder.


thinking out loud

Is "mine" objectively true in every case? In terms of phrasing, are "my car" and "your car" different?

What about mathematically? If any given person says "my car," are they referencing the same vehicle?

Exercise (PageIndex{2}): Compound Statements

Let (p) be the statement "some people are mortal," and (q) be the statement "All people can reason." State, in clear English, the following cases:

  1. ( eg p)
  2. ( eg q)
  3. (p wedge q)
  4. ( eg (p wedge q))

All people are not mortal, Some people can't reason, Some people are mortal and all people can reason, All people are not mortal or some people can't reason.

Exercise (PageIndex{3}): Truth Tables

Construct truth tables for the following statements:

  1. ( eg p wedge eg q)
(p)(q)( eg p)( eg p wedge eg q)( eg q)

2. ( eg q o p)

(p)(q)( eg q)( eg q o p)

3. ((p o q) wedge (q o p))

(p)(q)( p o q)((p o q) wedge (q o p))(q o p)

Exercise (PageIndex{4}): Intuitive Reasoning

Suppose you throw four darts at a dartboard. The board has four concentric sections:

  • A bull's eye, worth 10 points
  • The section nearest the middle, worth 8 points
  • The middle section, worth 6 points
  • The outer section, worth 4 points

Supposing that all four darts thrown hit the board, what kinds of scores are possible? What kinds of scores are impossible?

Exercise (PageIndex{5}): Negation of Statements with Quantifiers

Find the negation of ( forall x , exists y , s.t. , x-y=2.)


( forall y , exists x , s.t. , x-y e 2.)

Exercise (PageIndex{6}): Logical Equivalency

Prove or disprove the following statement: The expressions ((p vee q ) o r) and ( (p o r)wedge( q o r)) are logically equivalent.


The expressions ((p vee q ) o r) and ( (p o r)wedge( q o r)) are logically equivalent.

Exercise (PageIndex{7}): True or False

Assess whether each of the following statement is true or false and justify your answer.

  1. (7) is an integer and (-7>3.)
  2. ((-5)(-2) geq -10.)
  3. If ((4)(5)=10) then (frac{10}{4}=5.)
  4. If (8<5) then ( 8=5.)

F, T, T, T.

Exercise (PageIndex{8}): Tautology

Prove or disprove: for any mathematical statements (p,q) and (r), ((p leftrightarrow (( eg q) wedge( eg r))) o ( eg(q wedge r) o p)) is a tautology.

Exercise (PageIndex{9}): Logical Equivalents

Let (p) and (q) be mathematical statements. Then show that the following statements are true:

  1. (p vee q equiv q vee p), and (p wedge q equiv qwedge p).
  2. ( eg((p vee q)equiv eg p wedge eg q ) and ( eg((p wedge q) equiv eg p vee eg q )
  3. (p ightarrow q equiv eg q ightarrow eg p)
  4. (p ightarrow q equiv eg p vee q)
  5. ( eg ( eg p) equiv p)
  6. (p leftrightarrow q equiv ((p ightarrow q) wedge ( q ightarrow p))


(p)(q)(p vee q)(q vee p)

Exercise (PageIndex{10}): Reasoning

Determine whether the following arguments are valid or invalid:

  1. All polygons have angles.

A circle has no angle.

A circle is not a polygon.

  1. If you don’t work hard then you won’t succeed.

You work hard.

Therefore, you will succeed.

  1. If you make an A on the midterm, you won’t have to take the final.

Jose did not take the final.

Therefore, Jose made an A on the midterm.

  1. If a number is divisible by 8 then it is divisible by 4.

X is not divisible by 8.

Therefore x is not divisible by 4.

  1. If I am rich, I would buy a cabin.

I am not rich.

Therefore I have not bought a cabin.

6. If p is a prime number larger than 2 then p is odd.

p is odd.

Therefore p is a prime number.

1.E: Basic Language of Mathematics (Exercises)

This is a collection of programming exercises for or learners interested in language rather than math.

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Select Grade 7 ELA Worksheets by Topic

Explore 100+ Seventh Grade Language Arts Worksheets

Prepare for the nitty-gritty of adverbial phrases as you mine out and underline the group of words in each sentence that function as an adjective and describe the noun.

Hold an impromptu personification session with this English worksheet pdf, where the grade 7 learners choose a word from the box attributing human qualities to non-human things and complete each personification.

Take a walk back in history as you read this blast from the past. Write an interesting paragraph about putting a time capsule together capturing and preserving your favorite memories for your future generations.

Carve time out for this printable language arts worksheet that has an underlined clause in each sentence. Let grade 7 learners decide whether the clause is dependent or independent and write in the space provided.

This is the best place to begin collocating words. Encourage learners to sort the words in the box based on their collocations with the words "do", "make", and "raise". Frame sentences using any five collocations.

Scout out and underline the modifier that is separated from the word it describes in each sentence. Rewrite the sentences placing it correctly and complete the 7th grade English worksheet pdf.

Maximize your skills in detecting the active and passive voice using the hints: a sentence with an active voice has the subject at the beginning, while the word "by" signals passive voice.

Pull out this printable grade 7 ELA worksheet when the I'm-bored blues start to strike. Identify the sound words or onomatopoeia from the pictures. Hunt them in the word search grid and circle them.

Build on what learners already know about the Portuguese explorer, and the difficulties he encountered during his voyage as they read the passage, citing textual evidence, and analyzing connections.

Fast-track your compound direct object identification skills with this 7th grade language arts worksheet pdf. Spot the compound direct object in the sentence that receives the action of the verb and write it.

Do you often get balled up in using the words vain, vein, and vane? Learn their correct usage and complete the sentences, choosing between these confused words that have a world of difference in their meanings.

Put your inferring skills to good use, while reading each pair of sentences that describe the character explicitly or implicitly in this grade 7 printable ELA worksheet. Write "D" for direct and "I" for indirect characterization.

Change imperative sentences from direct to indirect speech, replacing the reporting verb with words like requested, ordered, suggested, and using the structure: reporting verb + noun / pronoun + to infinitive.

Thin as a toothpick, or pea-sized brains replete with such hyperboles, this grade 7 language arts worksheet pdf gets learners figuring out if there is an extreme exaggeration in a sentence and check the option.

Brainstorm learners on any important topic, and visually categorize and analyze all the possible causes and effects of the problem stated using this fishbone diagram template.

The Leveled Daily Curriculum has been growing over the past few years. There is Language Arts, Math, Science, Social Studies, and Functional Literacy. Level 0.5 and Level 4 has been added to LA and Math. I love these resources because it includes.

One of my favorite parts of the Leveled Daily Work is that is sets up a complete station of the classroom in one quick setup. You can print the daily work sheets or post them on a smartboard or iPad and photocopy the student response pages and you are.

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Types of Sets

We learned how to write sets using roster notation, as shown in examples 1 and 2 below.

Example 1: Let R be the set of all vowels in the English alphabet. Describe this set using roster notation.

Example 2: Let S be the set of all letters in the English alphabet. Describe this set using roster notation.

In example 2, there are 26 elements in set S. It would be easier to use a shortcut to list this set:

Example 2: Let S be the set of all letters in the English alphabet. Describe this set using roster notation.

The three dots are called an ellipsis. We use an ellipsis in the middle of a set as a shortcut for listing many elements. Note that the number of elements in set R and set S is countable, So each of these sets is a finite set. A finite set has a finite number of elements. Let's examine another type of set:

Example 3: Let T be the set of all whole numbers.

In example 3, we used an ellipsis at the end of the list to indicate that the set goes on forever. Set T is an infinite set. An infinite set is a set with an infinite number of elements. It is not possible to explicitly list out all the elements of an infinite set. Let's look at some more examples of finite and infinite sets.

Description Roster Notation Description Roster Notation
A = A = <1, 2, 3, . 97, 98, 99> W = W = <0, 2, 4, 6, 8, . >
B = B = X = X = . >
C = C = Y = Y = <2, 3, 5, 7, 11, . >

The ellipsis makes it easier to list both finite and infinite sets with roster notation. There are some sets that do not contain any elements at all, as shown below.

Example 4: Let D be the set of all weeks with 8 days.

We call a set with no elements the null or empty set. It is represented by the symbol < >or Ø . So D = <> or D = Ø. Let's look at some more examples of empty sets.

Empty (Null) Sets
Description Notation
The set of dogs with sixteen legs. X = <>
The set of computers that are both on and off. Y = <>
The set of triangles with 4 sides. Z = <>
The set of months with 32 days. D = Ø
The set of bicycles with no wheels. E = Ø
The set of whole numbers that are odd and even. F = Ø

Summary: An ellipsis is a shortcut used when listing sets with roster notation. A finite set has a countable number of elements: An infinite has an infinite number of elements, such as the set of whole numbers, which goes on forever. We call a set with no elements the null or empty set.

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The levels for ABE are beginning literacy, beginning basic education, low and high intermediate basic education, and low and high adult secondary education. Each ABE level measures basic reading, writing, numeracy, and functional and workplace skills that can be expected from a person functioning at that level.

Click on picture to view a small preview of each activity. Click on Doc or PDF to download worksheets in preferred format.

(1) (2) (3) (4)

(1) Category “grid” or squares 1: Doc PDF ( 2 ) Category “grid” or squares 2: Doc PDF (3) Category Tic Tac Toe – Connect 3 – Basic Doc PDF (4) Category Tic Tac Toe – Connect 3 Elementary Doc PDF (4) Category Tic Tac Toe – Connect 4 Higher Level Doc PDF (5) Tic Tac Toe Template 16 squares Doc PDF (6) Tic Tac Toe Template 25 squares Doc PDF

Reasons to Teach Sight Words.

In many schools students are expected to be able to read simple material by the end of the second grade. One of the most important goals in teaching young students to read is making sure they are completely proficient with Sight Words.

There are two additional reasons why it is important to give Sight Words an extra priority. Firstly, phonetic analysis can't be applied to many of these words. Secondly, quite a few of them cannot be taught through pictures (e.g. "if", "soon", "but", etc.).

Even though it may take a considerable effort for children to learn the entire Dolch Word List, it is well worth it. Having the ability to recognize these words can dramatically increase confidence and improve reading proficiency of the beginning reader.

Because complete fluency with Sight Words is the foundation of literacy, a variety of techniques are used to teach them to children. Repetition and practice are very important in making Sight Words recognition automatic. For example, there are numerous group activities that help in teaching Dolch Word List in elementary school setting. These activities are often supplemented by interactive games that involve flashcards and computer software.

Clearly, exercises and games like the ones presented on this website can be most valuable when used as a part of a comprehensive reading education program that employs a variety of materials and activities.

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