Can, Could, Be able to

Can and could are modal auxiliary verbs. Be able to uses the verb «to
be» as a main verb. It is not an auxiliary verb, but we look at it here
for convenience.

Can

Can is an auxiliary verb, a modal auxiliary verb. We use «can» to:

talk about possibility and ability

make requests

ask for or give permission

Structure of Can

subject + can + main verb

The main verb is always the bare infinitive (infinitive without «to»).

  subject auxiliary verb main verb  

+ I can play tennis.

— He cannot play tennis.

can’t

? Can you play tennis?

Notice that:

Can is invariable. There is only one form of can.

The main verb is always the bare infinitive.

Use of Can

can: Possibility and Ability

We use can to talk about what is possible, what we are able or free to
do:

She can drive a car.

John can speak Spanish.

I cannot hear you. (I can’t hear you.)

Can you hear me?

Normally, we use can for the present. But it is possible to use can when
we make present decisions about future ability.

Can you help me with my homework? (present)

Sorry. I’m busy today. But I can help you tomorrow. (future)

can: Requests and Orders

We often use can in a question to ask somebody to do something. This is
not a real question — we do not really want to know if the person is
able to do something, we want them to do it! The use of can in this way
is informal (mainly between friends and family):

Can you make a cup of coffee, please.

Can you put the TV on.

Can you come here a minute.

Can you be quiet!

can: Permission

We sometimes use can to ask or give permission for something:

Can I smoke in this room?

You can’t smoke here, but you can smoke in the garden.

(Note that we also use could, may, might for permission. The use of can
for permission is informal.)

Could

Could is an auxiliary verb, a modal auxiliary verb. We use «could» to:

talk about past possibility or ability

make requests

Structure of Could

subject + could + main verb

The main verb is always the bare infinitive (infinitive without «to»).

  subject auxiliary verb main verb  

+ My grandmother could speak Japanese.

— She could not speak Chinese.

couldn’t

? Could your grandmother speak Japanese?

Notice that:

Could is invariable. There is only one form of could.

The main verb is always the bare infinitive.

Use of Could

could: Past Possibility or Ability

We use could to talk about what was possible in the past, what we were
able or free to do:

I could swim when I was 5 years old.

My grandmother could speak seven languages.

When we arrived home, we could not open the door. (…couldn’t open the
door.)

Could you understand what he was saying?

We use could (positive) and couldn’t (negative) for general ability in
the past. But when we talk about one special occasion in the past, we
use be able (positive) and couldn’t (negative). Look at these examples:

  Past

General Specific Occasion

+ My grandmother could speak Spanish. A man fell into the river
yesterday. The police were able to save him.

— My grandmother couldn’t speak Spanish. A man fell into the river
yesterday. The police couldn’t save him.

could: Requests

We often use could in a question to ask somebody to do something. The
use of could in this way is fairly polite (formal):

Could you tell me where the bank is, please?

Could you send me a catalogue, please?

Be able to

Although we look at be able to here, it is not a modal verb. It is
simply the verb «to be» plus an adjective (able) followed by the
infinitive. We look at «be able to» here because we sometimes use it
instead of «can» and «could». We use «be able to»:

to talk about ability

Structure of Be able to

The structure of be able to is:

subject + be + able + infinitive

  subject be

main verb able

adjective infinitive

+ I am able to drive.

— She is not able to drive.

isn’t

? Are you able to drive?

Notice that be able to is possible in all tenses, for example:

I was able to drive…

I will be able to drive…

I have been able to drive…

Notice too that be able to has an infinitive form:

I would like to be able to speak Chinese.

Use of Be able to

be able to: ability

We use be able to to express ability. «Able» is an adjective meaning:
having the power, skill or means to do something. If we say «I am able
to swim», it is like saying «I can swim». We sometimes use «be able to»
instead of «can» or «could» for ability. «Be able to» is possible in all
tenses—but «can» is possible only in the present and «could» is possible
only in the past for ability. In addition, «can» and «could» have no
infinitive form. So we use «be able to» when we want to use other tenses
or the infinitive. Look at these examples:

I have been able to swim since I was five. (present perfect)

You will be able to speak perfect English very soon. (future simple)

I would like to be able to fly an airplane. (infinitive)

Have To (objective obligation)

We often use have to to say that something is obligatory, for example:

Children have to go to school.

Structure of Have To

«Have to» is often grouped with modal auxiliary verbs for convenience,
but in fact it is not a modal verb. It is not even an auxiliary verb. In
the «have to» structure, «have» is a main verb. The structure is:

subject + auxiliary verb + have + infinitive (with «to»)

Look at these examples in the simple tense:

  subject auxiliary verb main verb «have» infinitive (with «to»)  

+ She   has to work.  

— I do not have to see the doctor.

? Did you have to go to school?

Use of Have To

In general, «have to» expresses impersonal obligation. The subject of
«have to» is obliged or forced to act by a separate, external power (for
example, the Law or school rules). «Have to» is objective. Look at these
examples:

In France, you have to drive on the right.

In England, most schoolchildren have to wear a uniform.

John has to wear a tie at work.

In each of the above cases, the obligation is not the subject’s opinion
or idea. The obligation is imposed from outside.

We can use «have to» in all tenses, and also with modal auxiliaries. We
conjugate it just like any other main verb. Here are some examples:

  subject auxiliary verb main verb «have» infinitive  

past simple I   had to work yesterday.

present simple I   have to work today.

future simple I will have to work tomorrow.

present continuous She is having to wait.  

present perfect We have had to change the time.

modal (may) They may have to do it again.

Must (subjective obligation)

We often use must to say that something is essential or necessary, for
example:

I must go.

Structure of Must

«Must» is a modal auxiliary verb. It is followed by a main verb. The
structure is:

subject + must + main verb

The main verb is the base verb (infinitive without «to»).

Look at these examples:

subject auxiliary verb

«must» main verb  

I must go home.

You must visit us.

We must stop now.

 

Use of Must

In general, «must» expresses personal obligation. «Must» expresses what
the speaker thinks is necessary. «Must» is subjective. Look at these
examples:

I must stop smoking.

You must visit us soon.

He must work harder.

In each of the above cases, the «obligation» is the opinion or idea of
the person speaking. In fact, it is not a real obligation. It is not
imposed from outside.

We can use «must» to talk about the present or the future. Look at these
examples:

I must go now. (present)

I must call my mother tomorrow. (future)

There is no past tense for «must». We use «have to» to talk about the
past.

Must Not (prohibition)

We use must not to say that something is not permitted or allowed, for
example:

Passengers must not talk to the driver.

Structure of Must Not

«Must» is an auxiliary verb. It is followed by a main verb. The
structure for «Must Not» is:

Subject + «Must Not» + Main Verb

The Main Verb is the base verb (infinitive without «to»).

«Must Not» is often contracted to «mustn’t».

Look at these examples:

subject auxiliary «Must» + «Not» main verb  

I mustn’t forget my keys.

You mustn’t disturb him.

Students must not be late.

NB: like all auxiliary verbs, «must» cannot be followed by an
infinitive. So, we say:

You mustn’t arrive late. (not You mustn’t to arrive late.)

Use of Must Not

«Must Not» expresses prohibition — something that is not permitted, not
allowed. The prohibition can be subjective (the speaker’s opinion) or
objective (a real law or rule). Look at these examples:

I mustn’t eat so much sugar. (subjective)

You mustn’t watch so much television. (subjective)

Students must not leave bicycles here. (objective)

Policemen must not drink on duty. (objective)

We use «Must Not» to talk about the present or the future:

Visitors must not smoke. (present)

I mustn’t forget Tara’s birthday. (future)

We cannot use «Must Not» for the past. We use another structure to talk
about the past, for example:

We were not allowed to enter.

I couldn’t park outside the shop.

Shall and Will

People may sometimes tell you that there is no difference between shall
and will, or even that today nobody uses shall (except in offers such as
«Shall I call a taxi?»). This is not really true. The difference between
shall and will is often hidden by the fact that we usually contract them
in speaking with ‘ll. But the difference does exist.

The truth is that there are two conjugations for the verb will:

1st Conjugation (objective, simple statement of fact)

  Person Verb Example Contraction

Singular I shall I shall be in London tomorrow. I’ll

you will You will see a large building on the left. You’ll

he, she, it will He will be wearing blue. He’ll

Plural we shall We shall not be there when you arrive. We shan’t

you will You will find his office on the 7th floor. You’ll

they will They will arrive late. They’ll

 

2nd Conjugation (subjective, strong assertion, promise or command)

  Person Verb Example Contraction

Singular I will I will do everything possible to help. I’ll

you shall You shall be sorry for this. You’ll

he, she, it shall It shall be done. It’ll

Plural we will We will not interfere. We won’t

you shall You shall do as you’re told. You’ll

they shall They shall give one month’s notice. They’ll

It is true that this difference is not universally recognized. However,
let those who make assertions such as «Americans never use ‘shall'»
peruse a good American English dictionary, or many American legal
documents, which often contain phrases such as:

Each party shall give one month’s notice in writing in the event of
termination.

Note that exactly the same rule applies in the case of should and would.
It is perfectly normal, and somewhat more elegant, to write, for
example:

I should be grateful if you would kindly send me your latest catalogue.

Ten sentences:

Children have to go to school.

I must go to the university.

People mustn’t drive a car when they drink alcohol.

I needn’t do math today, I can do it later.

I should study harder before exams.

Elephants and mice can’t fly.

I could play snooker much better two years ago than I can now.

I can’t have made a mistake in my calculations because I used a
calculator.

Can you run 100 meters in 5.5 seconds? 10)

Students mustn’t eat or drink during the lection.

Texts:

Combinatorial mathematics.

Specialists in a broad range of fields have to deal with problems that
involve combinations made up of letters, numbers or any other objects.

The field of mathematics that studies problems of how many different
combinations can be built out of a specific number of objects is called
combinatorial mathematics (combinatorics).

This branch of mathematics has its origin in the 16th century, in the
gambling games that played such a large part in high society in those
times. These games gave the initial impetus to develop combinatorial
mathematics and the theory of probability.

Italian and French mathematicians were the first to enumerate the
various combinations achieved in games of dice. Further advances in the
theory of combinations were connected with the names of German
scientists.

In recent years combinatorial mathematics has seen extensive
developments associated with grater interest in problems of discrete
mathematics. Combinatorial methods can be employed in solving transport
problems, in particular scheduling; the scheduling of production
facilities and of the sale of goods. Links have been established between
combinatorics and problems of linear programming, statistics, etc.
Combinatorial methods are used in coding and decoding and in the
solution of other problems of information theory.

The combinatorial approach also plays a significant role in purely
mathematical problems such as the theory of groups and their
representations, in the study of the main principles of geometry, some
branches of algebra, etc.

Probability.

Probability is a mathematical expression of the likelihood of an event.
Every probability is a fraction. The largest probability can be 1. The
smallest probability can be is 0, meaning that it’s something that
cannot happen. You can find the probability that something will not
happen by subtracting the probability that it will happen from 1. For
example, if the weatherman tells you that there is a 0.3 probability of
rain today, then there must be a 0.7 probability that it won’t rain.

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