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