Is Statement And Negation Of Statement Same Thing?

Condider the following two statement

\[( \exists x \in R)\ (\forall y\in s)\ (y < x)\\ ( \exists x \in R) (\forall y\in s)\ ( y > x) \\ (\exists r\in R)\ (\forall y\in s)\ (|y| < r)\]

Determine which of these statements is true for each of the following choice of S

a (s) = [-3 , 10) (b) S = Q

Solution:

Here

\[( \exists x \in R) (\forall y\in s) (y < x)\\ ( \exists x \in R) (\forall y\in s) ( y > x) \\ (\exists r\in R) (\forall y\in s)(|y| < r)\]

a. All True b. All false

Write converse inverse and contrapositive of the statement. “ if x < 0 then \(x^2 – x > 0\) “, Also write the negation of each statement.

Solution:

Given statement is

\[' if\ x < 0\ then\ x^2 – x > 0 '\]

Converse:

\[if\ x^2 – x > 0\ then\ x < 0.\]

Nagation:

\[x^2 – x > 0\ and\ x > 0\]

Inverse

\[if\ x > 0\ then\ , x^2 – x < 0.\]

Negation:

\[x > 0\ and\ x^2 – x > 0\]

Contrapositive:

\[if\ x^2 – x < 0\ then\ x > 0\]

Negation:

\[x^2 – x < 0\ and\ x < 0\]