`star` Special Words/Phrases

`star` The word “And”

`star` The word “Or”

`star` Quantifiers

`star` The word “And”

`star` The word “Or”

`star` Quantifiers

`color{green} ✍️` Some of the connecting words which are found in compound statements like `"“And”, “Or”, "` etc. are often used in Mathematical Statements. These are called `"connectives."`

`color{green} ✍️` When we use these compound statements, it is necessary to understand the role of these words. We discuss this below.

`color{green} ✍️` When we use these compound statements, it is necessary to understand the role of these words. We discuss this below.

We have the following `color(red)("rules regarding the connective")` `color(blue)"“And”"`

`color{green} ✍️` `color(red)("Let us look at a compound statement")` with `"“And”."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" p: A point occupies a position and its location can be determined."`

The statement can be broken into two component statements as

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q: A point occupies a position."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" r: Its location can be determined."`

Here, we observe that `color(blue)("both statements are true.")`

`color{green} ✍️` `color(red)(" Let us look at another statement")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green} " p: 42 is divisible by 5, 6 and 7."`

This statement has following component statements

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " q: 42 is divisible by 5."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " r: 42 is divisible by 6."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " s: 42 is divisible by 7."`

Here, we know that the first is false while the other two are true.

`=>color(red)("Now, consider the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" p: A mixture of alcohol and water can be separated by chemical methods."`

This sentence cannot be considered as a compound statement with “And”. Here the word “And” refers to two things – alcohol and water.

`=>color(red)"This leads us to an important note"`.

`color(red)"Note"` Do not think that a statement with “And” is always a compound statement as shown in the above example.

Therefore, `color(blue)("the word “And” is not used as a conjunction.")`

`color(red)(1.) ` The compound statement with `color(blue)"“And”"` is true

`color(blue)("if all its component statements are true.")`

`color(red)(2.) ` The component statement with `color(blue)"“And”"` is false if any of its component statements is false (this includes the case that some of its component statements are false or all of its component statements are false).

`color(blue)("if all its component statements are true.")`

`color(red)(2.) ` The component statement with `color(blue)"“And”"` is false if any of its component statements is false (this includes the case that some of its component statements are false or all of its component statements are false).

`color{green} ✍️` `color(red)("Let us look at a compound statement")` with `"“And”."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" p: A point occupies a position and its location can be determined."`

The statement can be broken into two component statements as

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q: A point occupies a position."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" r: Its location can be determined."`

Here, we observe that `color(blue)("both statements are true.")`

`color{green} ✍️` `color(red)(" Let us look at another statement")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green} " p: 42 is divisible by 5, 6 and 7."`

This statement has following component statements

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " q: 42 is divisible by 5."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " r: 42 is divisible by 6."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange} " s: 42 is divisible by 7."`

Here, we know that the first is false while the other two are true.

`=>color(red)("Now, consider the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" p: A mixture of alcohol and water can be separated by chemical methods."`

This sentence cannot be considered as a compound statement with “And”. Here the word “And” refers to two things – alcohol and water.

`=>color(red)"This leads us to an important note"`.

`color(red)"Note"` Do not think that a statement with “And” is always a compound statement as shown in the above example.

Therefore, `color(blue)("the word “And” is not used as a conjunction.")`

Q 3141701623

Write the component statements of the following compound statements and check whether the compound statement is true or false.

(i) A line is straight and extends indefinitely in both directions.

(ii) 0 is less than every positive integer and every negative integer.

(iii) All living things have two legs and two eyes.

(i) A line is straight and extends indefinitely in both directions.

(ii) 0 is less than every positive integer and every negative integer.

(iii) All living things have two legs and two eyes.

(i) The component statements are

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: A line is straight."`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: A line extends indefinitely in both directions."`

Both these statements are true, therefore, the compound statement is true.

(ii) The component statements are

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: 0 is less than every positive integer."`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: 0 is less than every negative integer."`

The second statement is false. Therefore, the compound statement is false.

(iii) The two component statements are

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: All living things have two legs."`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: All living things have two eyes."`

Both these statements are false. Therefore, the compound statement is false.

Now, consider the following statement.

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: A mixture of alcohol and water can be separated by chemical methods."`

This sentence cannot be considered as a compound statement with “And”. Here the

word “And” refers to two things – alcohol and water.

This leads us to an important note.

`color(red)("Rule for the compound statement with ‘Or’"`

` => color(red)("Let us look at the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green)" p: Two lines in a plane either intersect at one point or they are parallel."`

`color(orange)("We know that this is a true statement.")`

This means that if two lines in a plane intersect, then they are not parallel.

Alternatively, if the two lines are not parallel, then they intersect at a point.

That is this statement is true in both the situations.

In order to understand statements with `color(blue)"“Or”"` we first notice that the word `"“Or”"` is used in two ways in English language.

` => color(red)("Let us look at the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green)" p: An ice cream or pepsi is available with a Thali in a restaurant."`

This means that a person who does not want ice cream can have a pepsi along with Thali or one does not want pepsi can have an ice cream along with Thali.

That is, who do not want a pepsi can have an ice cream. A person cannot have both ice cream and pepsi. This is called an `color(red)"exclusive “Or”."`

` => color(red)(" Here is another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: A student who has taken biology or chemistry can apply for M.Sc. microbiology programme."`

Here we mean that the students who have taken both biology and chemistry can apply for the microbiology programme, as well as the students who have taken only one of these subjects.

In this case, we are using `color(red)"inclusive “Or”."`

` => color(red)(" For example, consider the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: Two lines intersect at a point or they are parallel."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: Two lines intersect at a point."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " r: Two lines are parallel."`

Then, when `q` is true `r` is false and when `r` is true `q` is false. Therefore, the compound statement `p` is true.

` => color(red)(" Consider another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: 125 is a multiple of 7 or 8."`

Its component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: 125 is a multiple of 7."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " r: 125 is a multiple of 8."`

Both `q` and `r` are false. Therefore, the compound statement `p` is false.

` => color(red)(" Again, consider the following statement:")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: The school is closed, if there is a holiday or Sunday."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: School is closed if there is a holiday."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "r: School is closed if there is a Sunday."`

Both `q` and `r` are true, therefore, the compound statement is true.

` => color(red)("Consider another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: Mumbai is the capital of Kolkata or Karnataka."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "q: Mumbai is the capital of Kolkata."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "r: Mumbai is the capital of Karnataka."`

Both these statements are false. Therefore, the compound statement is false.

`color(red)(1.) ` A compound statement with an `color(blue)"‘Or’"` is true

`color(blue)("when one component statement is true or both the component statements are true.")`

`color(red)(2.) ` A compound statement with an `color(blue)"‘Or’"` is false `color(blue)("when both the component statements are false.")`

`color(blue)("when one component statement is true or both the component statements are true.")`

`color(red)(2.) ` A compound statement with an `color(blue)"‘Or’"` is false `color(blue)("when both the component statements are false.")`

` => color(red)("Let us look at the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green)" p: Two lines in a plane either intersect at one point or they are parallel."`

`color(orange)("We know that this is a true statement.")`

This means that if two lines in a plane intersect, then they are not parallel.

Alternatively, if the two lines are not parallel, then they intersect at a point.

That is this statement is true in both the situations.

In order to understand statements with `color(blue)"“Or”"` we first notice that the word `"“Or”"` is used in two ways in English language.

` => color(red)("Let us look at the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green)" p: An ice cream or pepsi is available with a Thali in a restaurant."`

This means that a person who does not want ice cream can have a pepsi along with Thali or one does not want pepsi can have an ice cream along with Thali.

That is, who do not want a pepsi can have an ice cream. A person cannot have both ice cream and pepsi. This is called an `color(red)"exclusive “Or”."`

` => color(red)(" Here is another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: A student who has taken biology or chemistry can apply for M.Sc. microbiology programme."`

Here we mean that the students who have taken both biology and chemistry can apply for the microbiology programme, as well as the students who have taken only one of these subjects.

In this case, we are using `color(red)"inclusive “Or”."`

` => color(red)(" For example, consider the following statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: Two lines intersect at a point or they are parallel."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: Two lines intersect at a point."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " r: Two lines are parallel."`

Then, when `q` is true `r` is false and when `r` is true `q` is false. Therefore, the compound statement `p` is true.

` => color(red)(" Consider another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: 125 is a multiple of 7 or 8."`

Its component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: 125 is a multiple of 7."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " r: 125 is a multiple of 8."`

Both `q` and `r` are false. Therefore, the compound statement `p` is false.

` => color(red)(" Again, consider the following statement:")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: The school is closed, if there is a holiday or Sunday."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " q: School is closed if there is a holiday."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "r: School is closed if there is a Sunday."`

Both `q` and `r` are true, therefore, the compound statement is true.

` => color(red)("Consider another statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(green) " p: Mumbai is the capital of Kolkata or Karnataka."`

The component statements are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "q: Mumbai is the capital of Kolkata."`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) "r: Mumbai is the capital of Karnataka."`

Both these statements are false. Therefore, the compound statement is false.

Q 3161701625

For each of the following statements, determine whether an inclusive “Or” or exclusive “Or” is used. Give reasons for your answer.

(i) To enter a country, you need a passport or a voter registration card.

(ii) The school is closed if it is a holiday or a Sunday.

(iii) Two lines intersect at a point or are parallel.

(iv) Students can take French or Sanskrit as their third language.

(i) To enter a country, you need a passport or a voter registration card.

(ii) The school is closed if it is a holiday or a Sunday.

(iii) Two lines intersect at a point or are parallel.

(iv) Students can take French or Sanskrit as their third language.

(i)Here “Or” is inclusive since a person can have both a passport and a voter registration card to enter a country.

(ii) Here also “Or” is inclusive since school is closed on holiday as well as on Sunday.

(iii) Here “Or” is exclusive because it is not possible for two lines to intersect and parallel together.

(iv) Here also “Or” is exclusive because a student cannot take both French and Sanskrit.

Q 3171701626

Identify the type of “Or” used in the following statements and check whether the statements are true or false:

(i) `sqrt2` is a rational number or an irrational number.

(ii) To enter into a public library children need an identity card from the school

or a letter from the school authorities.

(iii) A rectangle is a quadrilateral or a 5-sided polygon.

(i) `sqrt2` is a rational number or an irrational number.

(ii) To enter into a public library children need an identity card from the school

or a letter from the school authorities.

(iii) A rectangle is a quadrilateral or a 5-sided polygon.

(i)The component statements are

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sqrtp: 2 "is a rational number."`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sqrtq: 2 "is an irrational number"`

Here, we know that the first statement is false and the second is true and “Or” is

exclusive. Therefore, the compound statement is true.

(ii) The component statements are

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"p: To get into a public library children need an identity card."`

` \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: To get into a public library children need a letter from the school authorities."`

Children can enter the library if they have either of the two, an identity card or the

letter, as well as when they have both. Therefore, it is inclusive “Or” the compound

statement is also true when children have both the card and the letter.

(iii) Here “Or” is exclusive. When we look at the component statements, we get that

the statement is true.

Quantifiers are phrases like, `color(blue)"“There exists”"` and `color(blue)"“For all”."`

The words `color(blue)"“And”"` and `color(blue)"“Or”"` are `color(red)("called connectives")` and `color(blue)"“There exists”"` and `color(blue)"“For all”"` are `color(red)("called quantifiers.")`

Another phrase which appears in mathematical statements is “there exists”.

`=>color(red)("For example, consider the statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " p: There exists a rectangle whose all sides are equal."`

This means that there is atleast one rectangle whose all sides are equal.

A word closely connected with `"“there exists”"` is `"“for every”"` (or for all).

`=>color(red)("Consider a statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) " p: For every prime number p,"`` color(orange) (sqrt(p) "is an irrational number.")`

This means that if `S` denotes the set of all prime numbers, then for all the members `p` of the set `S, sqrt(p)` is an irrational number.

In general, a mathematical statement that says “for every” can be interpreted as saying that all the members of the given set S where the property applies must satisfy that property.

We should also observe that it is important to know precisely where in the sentence a given connecting word is introduced. For example, compare the following two sentences:

`1.` For every positive number x there exists a positive number y such that `y < x.`

`2. ` There exists a positive number y such that for every positive number x, we have `y < x.`

Although these statements may look similar, they do not say the same thing. As a matter of fact, (1) is true and (2) is false.

Thus, in order for a piece of mathematical writing to make sense, all of the symbols must be carefully introduced and each symbol must be introduced at precisely the right place – not too early and not too late.

The words `color(blue)"“And”"` and `color(blue)"“Or”"` are `color(red)("called connectives")` and `color(blue)"“There exists”"` and `color(blue)"“For all”"` are `color(red)("called quantifiers.")`

Another phrase which appears in mathematical statements is “there exists”.

`=>color(red)("For example, consider the statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " p: There exists a rectangle whose all sides are equal."`

This means that there is atleast one rectangle whose all sides are equal.

A word closely connected with `"“there exists”"` is `"“for every”"` (or for all).

`=>color(red)("Consider a statement.")`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) " p: For every prime number p,"`` color(orange) (sqrt(p) "is an irrational number.")`

This means that if `S` denotes the set of all prime numbers, then for all the members `p` of the set `S, sqrt(p)` is an irrational number.

In general, a mathematical statement that says “for every” can be interpreted as saying that all the members of the given set S where the property applies must satisfy that property.

We should also observe that it is important to know precisely where in the sentence a given connecting word is introduced. For example, compare the following two sentences:

`1.` For every positive number x there exists a positive number y such that `y < x.`

`2. ` There exists a positive number y such that for every positive number x, we have `y < x.`

Although these statements may look similar, they do not say the same thing. As a matter of fact, (1) is true and (2) is false.

Thus, in order for a piece of mathematical writing to make sense, all of the symbols must be carefully introduced and each symbol must be introduced at precisely the right place – not too early and not too late.