`color{green} ✍️` In this Section, we shall discuss the implications of `color(blue)"“if-then”, “only if”"` and `color(blue)"“if and only if ”."` The statements with `color(blue)"“if-then”"` are very common in mathematics.
`=>color(red)("For example, consider the statement.")`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" r: If you are born in some country, then you are a citizen of that country."`
When we look at this statement, we observe that it corresponds to two statements `p` and `q` given by
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : you are born in some country."`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q : you are citizen of that country."`
Then the sentence `color(blue)"“if p then q”"` says that in the event if `p` is true, then `q` must be true.
One of the most important facts about the sentence `"“if p then q”"` is that it does not say any thing (or places no demand) on `q` when `p` is false.
For example, if you are not born in the country, then you cannot say anything about `q.` To put it in other words” not happening of `p` has no effect on happening of `q.`
Another point to be noted for the statement `"“if p then q”"` is that the statement does not imply that `p` happens.
There are several ways of understanding “if p then q” statements.
`=>` We shall illustrate these ways in the context of the following statement.
` \ \ \ \ \ \ \ \ \ \ \ \ \color{green} " r: If a number is a multiple of 9, then it is a multiple of 3."`
Let p and q denote the statements
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : a number is a multiple of 9."`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q: a number is a multiple of 3."`
Then, if `p` then `q` is the same as the following:
`color(red)(1.) \ \p \ \ "implies" \ \ q` is denoted by `p ⇒ q.` The symbol `⇒` stands for implies. This says that a number is a multiple of 9 implies that it is a multiple of 3.
`color(red)(2.) \ \ p` is a sufficient condition for `q.`
This says that knowing that a number as a multiple of `9` is sufficient to conclude that it is a multiple of `3.`
`color(red)(3.) \ \p` only if `q.`
This says that a number is a multiple of `9` only if it is a multiple of `3.`
`color(red)(4.) \ \q` is a necessary condition for `p.`
This says that when a number is a multiple of `9,` it is necessarily a multiple of `3.`
`color(red)(5.) \ \∼q " implies" ∼p.`
This says that if a number is not a multiple of `3`, then it is not a multiple of `9.`
`color{green} ✍️` In this Section, we shall discuss the implications of `color(blue)"“if-then”, “only if”"` and `color(blue)"“if and only if ”."` The statements with `color(blue)"“if-then”"` are very common in mathematics.
`=>color(red)("For example, consider the statement.")`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{green}" r: If you are born in some country, then you are a citizen of that country."`
When we look at this statement, we observe that it corresponds to two statements `p` and `q` given by
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : you are born in some country."`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q : you are citizen of that country."`
Then the sentence `color(blue)"“if p then q”"` says that in the event if `p` is true, then `q` must be true.
One of the most important facts about the sentence `"“if p then q”"` is that it does not say any thing (or places no demand) on `q` when `p` is false.
For example, if you are not born in the country, then you cannot say anything about `q.` To put it in other words” not happening of `p` has no effect on happening of `q.`
Another point to be noted for the statement `"“if p then q”"` is that the statement does not imply that `p` happens.
There are several ways of understanding “if p then q” statements.
`=>` We shall illustrate these ways in the context of the following statement.
` \ \ \ \ \ \ \ \ \ \ \ \ \color{green} " r: If a number is a multiple of 9, then it is a multiple of 3."`
Let p and q denote the statements
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" p : a number is a multiple of 9."`
` \ \ \ \ \ \ \ \ \ \ \ \ \ color{orange}" q: a number is a multiple of 3."`
Then, if `p` then `q` is the same as the following:
`color(red)(1.) \ \p \ \ "implies" \ \ q` is denoted by `p ⇒ q.` The symbol `⇒` stands for implies. This says that a number is a multiple of 9 implies that it is a multiple of 3.
`color(red)(2.) \ \ p` is a sufficient condition for `q.`
This says that knowing that a number as a multiple of `9` is sufficient to conclude that it is a multiple of `3.`
`color(red)(3.) \ \p` only if `q.`
This says that a number is a multiple of `9` only if it is a multiple of `3.`
`color(red)(4.) \ \q` is a necessary condition for `p.`
This says that when a number is a multiple of `9,` it is necessarily a multiple of `3.`
`color(red)(5.) \ \∼q " implies" ∼p.`
This says that if a number is not a multiple of `3`, then it is not a multiple of `9.`