`\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} ✍️` 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
Check whether the following statement is true or not.