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Recall that a formula is a statement whose truth valuemay depend on the values of some variables. For example,

'$xle 5 land x> 3$'

is true for $x= 4$ and false for$x= 6$. Compare this with the statement

Quantifiers: worksheets pdf, handouts to print - quantity words. Printable exercises. Many - much - a lot of - few - little - less - fewer - more.

'For every $x$, $xle 5 land x>3$,'

which is definitely false and the statement

• Table of Contents. 1 Vocabulary: Likes and Dislikes in English. 1.1 Key verbs to Talk about Likes and Dislikes; 1.2 Action Verbs to Talk about Likes and Dislikes; 1.3 Grammar to Talk about Likes and Dislikes.
• Download Free PDF. English Grammar.pdf. English Grammar.pdf. Download Full PDF Package. A short summary of this.
• Quantifiers are a type of determiner which denote imprecise quantity. They modify nouns or pronouns. They differ from numbers or numerals which indicate precise quantity. The most common quantifiers used in English are: some / any, much, many, a lot, a few, several, enough.

'There exists an $x$ such that $xle 5 land x>3$,'

which is definitely true. The phrase 'for every $x$'(sometimes 'for all $x$') is calleda universal quantifier and is denoted by $forall x$. The phrase 'thereexists an $x$ such that' is called an existentialquantifier and is denotedby $exists x$. A formula that contains variables is not simplytrue or false unless each of these variables is bound by a quantifier. If a variable is notbound the truth of the formula is contingent on the value assigned tothe variablefrom the universe of discourse.

We were careful in section 1.1 to definethe truth values of compound statements precisely. We do the same for$forall x,P(x)$ and $exists x,P(x)$, though the intended meaningsof these are clear.

The Universal Quantifier

A sentence $forall x,P(x)$ is true if and only if $P(x)$ is true nomatter what value (from the universe of discourse) is substituted for $x$.

Example 1.2.1

$bullet$ $forall x (x^2ge 0)$,i.e., 'the square of any number is not negative.'

$bullet$ $forall x,forall y (x+y=y+x)$, i.e., the commutative law of addition.

$bullet$ $forall x,forall y,forall z ((x+y)+z=x+(y+z))$,i.e., the associative law of addition.

$square$

The 'all' form.The universal quantifier is frequently encountered in the following context:$$forall x (P(x)implies Q(x)),$$which may be read, 'All $x$ satisfying $P(x)$ also satisfy$Q(x)$.' Parentheses are crucial here; be sure you understand thedifference between the 'all' form and $forall x,P(x)impliesforall x,Q(x)$ and $(forall x,P(x))implies Q(x)$.

The latter formula might also be written as $forall x,P(x)impliesQ(x)$, which is to say that the universal quantifier has higherprecedence than the conditional; to avoid misunderstanding,it is best to include the parentheses. The meaning of this formulamight not be clear at first. The $x$ in $P(x)$ is bound by theuniversal quantifier, but the $x$ in $Q(x)$ is not. The formula$(forall x,P(x))implies Q(x)$ has the same meaning as $(forallx,P(x))implies Q(y)$, and its truth depends on the value assignedto the variable in $Q(cdot)$.

Example 1.2.2

$bullet$ $forall x$ ($x$ is a square $implies$ $x$ is a rectangle),i.e., 'all squares are rectangles.'

$bullet$ $forall x$ ($x$ lives in Walla Walla $implies$ $x$ lives in Washington), i.e., 'every person who lives in Walla Walla lives in Washington.'

$square$

This construction sometimes is used to express amathematical sentence of the form 'if this, then that,' with an'understood' quantifier.

Example 1.2.3

$bullet$ If we say, 'if $x$ is negative, so is its cube,' weusually mean 'every negative $x$ has a negative cube.' This shouldbe written symbolically as$forall x ((x< 0)implies (x^3< 0))$.

$bullet$ 'If two numbers have the same square, then they have the same absolute value' should be written as$forall x,forall y ((x^2=y^2)implies(vert xvert = vert yvert))$.

$bullet$ 'If $x=y$, then $x+z=y+z$' should be written as $forall x,forally,forall z ((x=y)implies (x+z=y+z))$.

$square$

If $S$ is a set, the sentence 'every $x$ in $S$ satisfies $P(x)$' iswritten formally as$$forall x ((xin S)implies P(x))$$ For clarity and brevity, this is usually written $forall x,{in},S,(P(x))$. To understand and manipulate the formula $forallx,{in},S, (P(x))$ properly, you will sometimes need to'unabbreviate' it, rewriting it as $forall x ((xin S)impliesP(x))$.

Example 1.2.4

$bullet$ $forall xin [0,1] (sqrt xge x)$stands for $forall x (xin [0,1]implies sqrt xge x).$

$bullet$ $forall x< 0 (vert xvert =-x)$ stands for $forall x (x< 0implies vert xvert = -x).$

$square$

The Existential Quantifier

A sentence $exists x,P(x)$ is true if and only if there is at leastone value of $x$ (from the universe of discourse) that makes $P(x)$ true.

Example 1.2.5

$bullet$ $exists x (x ge x^2)$is true since $x=0$ is a solution. There are many others.

$bullet$ $exists x,exists y (x^2+y^2=2xy)$ is true since$x=y=1$ is one of many solutions.

$square$

The 'some' form. The existentialquantifier is frequently encountered in the following context: $$exists x(P(x)land Q(x)),$$ which may be read, 'Some $x$ satisfying $P(x)$ alsosatisfies $Q(x)$.'

Example 1.2.6

$bullet$ $exists x, hbox{($x$ is a professor $land$ $x$ is a republican)}$, i.e., 'some professor is a republican.'

$bullet$ $exists x, hbox{($x$ is a prime number $land$ $x$ is even)}$, i.e.,'some prime number is even.'

$square$

It may at first seem that 'Some $x$ satisfying $P(x)$satisfies $Q(x)$' should be translated as$$exists x (P(x)implies Q(x)),$$like the universal quantifier. To see why this does not work,suppose $P(x)=hbox{'$x$ is an apple'}$ and $Q(x)=hbox{'$x$ is anorange.'}$ The sentence 'some apples are oranges' is certainlyfalse, but$$exists x (P(x)implies Q(x))$$is true. To see this suppose $x_0$ is some particular orange. Then$P(x_0)implies Q(x_0)$ evaluates to $hbox{F}implies hbox{T}$,which is T, and the existential quantifier is satisfied.

We use abbreviations of the 'some' form much like those for the'all' form.

Example 1.2.7

$bullet$ $exists x< 0 (x^2=1)$ stands for $exists x ((x< 0) land (x^2=1))$

$bullet$ $ exists xin [0,1] (2x^2+x =1)$ stands for $ exists x ((xin [0,1])land (2x^2+x=1))$$square$

If $forall$ corresponds to 'all' and $exists$ corresponds to 'some'do we need a third quantifier to correspond to 'none'? As the following shows, this is not necessary:

Example 1.2.8

$bullet$ 'No democrats are republicans,' can be written $forall x$ ($x$ is a democrat $implies$ $x$ is not a republican).

$bullet$ 'No triangles are rectangles,' can be written $forall x$ ($x$ is a triangle $implies$ $x$ is not a rectangle).

$square$

Clear Explanation About English Quantifiers Pdf Free

In general, the statement 'no $x$ satisfying $P(x)$ satisfies $Q(x)$' can be written $$forall x (P(x)implies lnot Q(x)).$$(You may wonder why we do not use $lnot exists x,(P(x)land Q(x))$. In fact, we could—it is equivalent to $forall x (P(x)implies lnot Q(x))$.)

Exercises 1.2

In these problems, assume the universe of discourse is thereal numbers.

Ex 1.2.1Express the following as formulas involving quantifiers:

a) Any number raised to the fourth power is non-negative.

b) Some number raised to the third power is negative.

c) The sine of an angle is always between $+1$ and $-1$.

d) The secant of an angle is never strictly between $+1$ and $-1$.

Ex 1.2.2Suppose $X$ and $Y$ are sets. Express the following as formulas involving quantifiers.

a) Every element of $X$ is an element of $Y$.

b) Some element of $X$ is an element of $Y$.

c) Some element of $X$ is not an element of $Y$.

d) No element of $X$ is an element of $Y$.

Ex 1.2.3Recall (from calculus) that a function $f$ is increasing if$$ forall a forall b (a< bimplies f(a)< f(b))$$Express the following definitions as formulas involving quantifiers:

a) $f$ is decreasing.

b) $f$ is constant.

c) $f$ has a zero.

Ex 1.2.4Express the following laws symbolically:

a) the commutative law of multiplication

b) the associative law of multiplication

c) the distributive law

Ex 1.2.5Are the following sentences true or false?

a) $forall x forall y (x< yimplies x^2< y^2)$

b) $forall x forall y forall zne 0 (xz=yzimplies x=y)$

c) $exists x< 0 exists y< 0 (x^2+xy+y^2=3)$

d) $exists x exists y exists z (x^2+y^2+z^2=2xy-2+2z)$

Ex 1.2.6Suppose $P(x)$ and $Q(y)$ are formulas.

a) Is $forall x forall y (P(x)implies Q(y))$equivalent to $forall x(P(x)) implies forall y(Q(y))$?

b) Is $exists x exists y (P(x)land Q(y))$equivalent to $exists x(P(x)) land exists y(Q(y))$?

Using quantifiers in English - a guide
Part 1

Click for :

Quantifiers of large quantity

Quantifiers of small quantity

Neutral and relative quantifiers

1. Large quantity quantifiers:

much, many, lots of, plenty of, numerous, a large number of, etc.

► Much and many :

Much is used with non-count nouns (always in the singular); many is used with ► count nouns in the plural. (See ►the difference between count nouns and non-count nouns).

Much and many in affirmative statements

In modern spoken English, Much, and to a lesser extent many are not often used as quantifiers in affirmative statements, unless introduced by an intensifier, notably so or too , or followed by of;
Remember : don't use much or many in affirmative statements, if you can avoid it. Though their use may be possible, it often sounds very formal, old-fashioned or strange in modern English.

Much and many in negative statements and questions

Much, and many are more commonly used in interrogative and negative contexts, and most particularly in the interrogative expressions How much and How many.

► Lots of, a lot of, plenty of, a large number of, numerous

These expressions are all more or less synonyms. In the list above, they are arranged in order of formality, going from the most informal (lots of) to the most formal (numerous). Informal language is more appropriate in dialogue, formal language in written documents.
For more on style, see ►styles of English .

► Much / many or Much of / many of ?

As quantifiers, much and many are not followed by of when they quantify a noun directly. However they must be followed by of if they come before a determiner such as an article, a possessive or a demonstrative. The same principle applies to few / few of (see below), some / some of, etc..

► Several and a number of

These imply 'more than one, but less than a lot'. They are not usually used in negative or interrogative structures, only in affirmative statements. For example

2. Small quantity quantifiers:

► few, a few, little, a little, not many, not much, a small number of, etc.

Except for not much or not many, these quantifiers are generally used in affirmative statements.

► Little, a little, not muchare used with non-count nouns (always in the singular)
Few, a few, not many are used with count nouns in the plural.

Few and little imply a quantity which is essentially small or smaller than expected.
A few and a little imply small quantity, but possibly more than expected

3. Neutral and relative quantifiers:

Neutral quantifiers do not indicate either a large quantity or a small quantity: they are not really concerned by actual quantity, only by relative quantity. They are dealt with in four different groups:

1. ► Some and any (see specific page)
2. ► Each and every (see specific page)
3. ► All and whole (see specific page)
4. Most, most of and enough - See below

Most, most of and enough

There are a couple of common quantifiers that express relative or proportional quantity.

► Most / most of

These imply more than half of, a majority of, or almost all . They do not mean the same as many / many of

► Enough

Enough implies a sufficient quantity; it is used in affirmations, negations and questions.

NOTE : do not confuse....

enough as a quantifier adjective preceding a noun, as in

Clear Explanation About English Quantifiers Pdf Format

with enough as an intensifier following an ►adjective, as in:
Click for more about ►enough :

4. Recapitulation: table of usage for common English quantifiers

Affirm­ative	Negative	Interrog­ative
Neutral	some, several, a number of, enough	any, enough	any, enough
Large quantity	numerous, plenty of, a lot of, lots of, too many	much, many, too many	much, many, too many
Small quantity	few / a few, Little / a little

► For quantifiers used as pronouns, see Indefinite pronouns, quantifiers & gender-neutral pronouns

4.1. QUANTIFIERS WITH OF... TAKE CARE:

Much of, many of, few of, a little of, plenty of, lots of, some of, a number of, none of, several of, etc.

When followed by of, some of these quantifiers MUST be followed by an article or other determiner; for others there is a choice (article or no article)

The rule....	... applies to
MUST be followed by an article or other determiner	all of, each of, some of, many of, much of, (a) few of, (a) little of, none of, several of, enough of,
MAY or MAY NOT be followed by an article or other determiner	plenty of, a lot of, lots of, a number of, a couple of,

Here are a few examples; most are right, some (in grey and barred out) are wrong.

Clear Explanation About English Quantifiers Pdf File

5. Few or a few, little or a little ?

The difference between the two expressions in each phrase is purely one of meaning, not of usage.

Clear Explaination About English Quantifiers Pdf

Clear Explanation About English Quantifiers Pdf Download

Without the article, few and little (used respectively with count nouns and non-count nouns) have the meaning of 'not much/ not many, and possibly less than one might hope for or expect'. These expressions have a negative value to them.
With the article, a few and a little have the meaning of 'at least some, perhaps more than one might expect' . These expressions have a positive value.
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