`color(red)(star)` Validating Statements

`color(red)(star)` By Contradiction

`color(red)(star)` By Contradiction

`\color{red} ✍️` In this Section, we will discuss when a statement is true. To answer this question, one must answer all the following questions.

`\color{red} ✍️` What does the statement mean `?` What would it mean to say that this statement is true and when this statement is not true?

The answer to these questions depend upon which of the special words and phrases `color{purple}("“and”, “or”,")` and which of the implications `color{purple}("“if and only”, “if-then”,")` and which of the quantifiers `color{purple}("“for every”, “there exists”,")` appear in the given statement.

`color(red)(=>"Here, we shall discuss some techniques to find when a statement is valid.")`

`color(green)ul"We shall list some general rules for checking whether a statement is true or not."`

`color{red}("Rule 1 : ")` If `p` and `q` are mathematical statements, then in order to show that the statement `color{navy}("“p and q” is true")`, the following steps are followed.

`color{green}("Step-1")` Show that the statement `p` is true.

`color{green}("Step-2")` Show that the statement `q` is true.

`color{red}("Rule 2 : ")` `color{blue}("Statements with “Or”")`

If `p` and `q` are mathematical statements , then in order to show that the statement `color{navy}("“p or q” is true")`, one must consider the following.

`color{green}("Case 1-")` By assuming that `p` is false, show that `q` must be true.

`color{green}("Case 2-")` By assuming that `q` is false, show that `p` must be true.

`color{red}("Rule 3 : ")` `color{blue}("Statements with “If-then”")`

In order to prove the statement `color{navy}("“if p then q”")` we need to show that any one of the following case is true.

`color{green}("Case 1-: ")` By assuming that `p` is true, prove that `q` must be true.(Direct method)

`color{green}("Case 2 - ")` By assuming that `q` is false, prove that `p` must be false.(Contrapositive method)

`color{red}("Rule 4 : ")` `color{blue}("Statements with “if and only if ” ")`

In order to prove the statement `color{navy}("“p if and only if q”",)` we need to show.

`(i)` If `p` is true, then `q` is true and

`(ii)` If `q` is true, then p is true

`\color{red} ✍️` What does the statement mean `?` What would it mean to say that this statement is true and when this statement is not true?

The answer to these questions depend upon which of the special words and phrases `color{purple}("“and”, “or”,")` and which of the implications `color{purple}("“if and only”, “if-then”,")` and which of the quantifiers `color{purple}("“for every”, “there exists”,")` appear in the given statement.

`color(red)(=>"Here, we shall discuss some techniques to find when a statement is valid.")`

`color(green)ul"We shall list some general rules for checking whether a statement is true or not."`

`color{red}("Rule 1 : ")` If `p` and `q` are mathematical statements, then in order to show that the statement `color{navy}("“p and q” is true")`, the following steps are followed.

`color{green}("Step-1")` Show that the statement `p` is true.

`color{green}("Step-2")` Show that the statement `q` is true.

`color{red}("Rule 2 : ")` `color{blue}("Statements with “Or”")`

If `p` and `q` are mathematical statements , then in order to show that the statement `color{navy}("“p or q” is true")`, one must consider the following.

`color{green}("Case 1-")` By assuming that `p` is false, show that `q` must be true.

`color{green}("Case 2-")` By assuming that `q` is false, show that `p` must be true.

`color{red}("Rule 3 : ")` `color{blue}("Statements with “If-then”")`

In order to prove the statement `color{navy}("“if p then q”")` we need to show that any one of the following case is true.

`color{green}("Case 1-: ")` By assuming that `p` is true, prove that `q` must be true.(Direct method)

`color{green}("Case 2 - ")` By assuming that `q` is false, prove that `p` must be false.(Contrapositive method)

`color{red}("Rule 4 : ")` `color{blue}("Statements with “if and only if ” ")`

In order to prove the statement `color{navy}("“p if and only if q”",)` we need to show.

`(i)` If `p` is true, then `q` is true and

`(ii)` If `q` is true, then p is true

Q 3161801725

Check whether the following statement is true or not.

If `x, y ∈ Z` are such that x and y are odd, then `xy` is odd.

If `x, y ∈ Z` are such that x and y are odd, then `xy` is odd.

Let `p : x, y ∈ Z` such that `x` and `y` are odd

`q : xy` is odd

To check the validity of the given statement, we apply Case 1 of Rule 3. That is

assume that if p is true, then q is true.

p is true means that `x` and `y` are odd integers. Then

`x = 2m + 1,` for some integer `m. y = 2n + 1,` for some integer n. Thus

`xy = (2m + 1) (2n + 1)`

`= 2(2mn + m + n) + 1`

This shows that `xy` is odd. Therefore, the given statement is true.

Suppose we want to check this by using Case 2 of Rule 3, then we will proceed as follows.

We assume that q is not true. This implies that we need to consider the negation of the statement q. This gives the statement `∼q` : Product `xy` is even.

This is possible only if either `x` or `y` is even. This shows that p is not true. Thus we

This is possible only if either x or y is even. This shows that p is not true. Thus we have shown that

`"∼q ⇒ ∼p"`

`\color{red} ✍️` `color(blue)"Here to check whether a statement p is true"`,

`color(green)("we assume that p")` is not true i.e. `color{green}(∼p" is true")`. Then,

we arrive at some result which contradicts our assumption.

Therefore, `color(navy)"we conclude that p is true."`

`color(green)("we assume that p")` is not true i.e. `color{green}(∼p" is true")`. Then,

we arrive at some result which contradicts our assumption.

Therefore, `color(navy)"we conclude that p is true."`