Mathematical Reasoning

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Let p: If x is an integer and x² is even, then x is also even.

Let q: x be an integer and x² be even

Let r: x is even

We have to prove, using the method of contra positive whether $q\Rightarrow r$ is true.

[i.e., its contra positive $\sim r\Rightarrow \sim q$ is true]

Let $\sim r$ be true i.e., r be false

Let us assume that x is not even (integer) i.e., x is odd.

$\therefore x=2m+1$ where m is an integer

$\therefore x^2{\left(2m+1\right)}^2=4m+1+4m$

$=4m^2+4m+1$

$=2\left(2m^2+2m+1\right)$ $=2t+1t=2m^2+2m$ is an integer.

$\Rightarrow$ x² is also odd $\Rightarrow$ x is not even

$\Rightarrow q\text{\hspace{0.17em}isfalse}\left[\text{\hspace{0.17em}Bydefof}q\right]\Rightarrow \sim q\text{\hspace{0.17em}istrue}$

$\therefore \sim r\Rightarrow \sim q$ is true

Thus, by the method of contrapositive $q\Rightarrow r$ is true

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Let p: If x is an integer and x² is even, then x is also even.

Let q: x be an integer and x² be even

Let r: x is even

We have to prove, using the method of contra positive whether $q\Rightarrow r$ is true.

[i.e., its contra positive $\sim r\Rightarrow \sim q$ is true]

Let $\sim r$ be true i.e., r be false

Let us assume that x is not even (integer) i.e., x is odd.

$\therefore x=2m+1$ where m is an integer

$\therefore x^2{\left(2m+1\right)}^2=4m+1+4m$

$=4m^2+4m+1$

$=2\left(2m^2+2m+1\right)$ $=2t+1t=2m^2+2m$ is an integer.

$\Rightarrow$ x² is also odd $\Rightarrow$ x is not even

$\Rightarrow q\text{\hspace{0.17em}isfalse}\left[\text{\hspace{0.17em}Bydefof}q\right]\Rightarrow \sim q\text{\hspace{0.17em}istrue}$

$\therefore \sim r\Rightarrow \sim q$ is true

Thus, by the method of contrapositive $q\Rightarrow r$ is true

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Let a = 2, b = –2 be two real numbers.

Clearly, a² = b² (=4) but $a\ne b.$

Thus, the given statement is not true.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(a) (i) Contra positive

(ii) Converse

(b) (i) Contra positive

(ii) Converse

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) If you get a job, then your credentials are good.

(ii) If the Banana trees stay warm for a month, then the trees will bloom.

(iii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) If you want to score an A+ in the class, then you do all the exercises in the book.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Five different ways are–

(i) A natural number is odd implies that its square is odd.

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd it is necessary that its square is odd.

(iv) For the square of a natural number to be odd it is sufficient that the number is odd.

(v) If the square of a natural number is not odd, then the natural number is not odd.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

(i) The 'or' in this statement is conclusive because it is not possible for the sun to rise and the moon to set together.

(ii) The 'or' in this statement is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) The 'or' in this statement is exclusive since all integers cannot be both positive and negative.

This is a  Mathematical Reasoning Solutions Type Questions as classified in NCERT Exemplar

Negation of statement (i) is:

There exists real numbers x and y for which x + y = y + x.

Now, this statement is not same as statement (ii).

Therefore, the given statements are not negation of each other.