Proofs In Mathematics A1

State whether the following statements are always true, always false or ambiguous. Justify your answers.

State whether the following statements are true or false. Justify your answers.

Let $a$ and $b$ be real numbers such that $ab \neq 0$. Then which of the following statements are true? Justify your answers.

Restate the following statements with appropriate conditions, so that they become true.

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A?

Given that the product of two rational numbers is rational, and suppose $a$ and $b$ are rationals, what can you conclude about $ab$?

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and $\sqrt{17}$ is irrational, what can we conclude about the decimal expansion of $\sqrt{17}$?

Given that $y = x^2 + 6$ and $x = -1$, what can we conclude about the value of $y$?

Given that PQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

Given that $\sqrt{p}$ is irrational for all primes $p$ and also suppose that 3721 is a prime. Can you conclude that $\sqrt{3721}$ is an irrational number? Is your conclusion correct? Why or why not?

Prove that the sum of two consecutive odd numbers is divisible by 4.

Take two consecutive odd numbers. Find the sum of their squares, and then add 6 to the result. Prove that the new number is always divisible by 8.

If $p \geq 5$ is a prime number, show that $p^2 + 2$ is divisible by 3.

Let $x$ and $y$ be rational numbers. Show that $xy$ is a rational number.

If $a$ and $b$ are positive integers, then you know that $a = bq + r, 0 \leq r < b$, where $q$ is a whole number. Prove that HCF $(a, b) = \text{HCF}(b, r)$.

A line parallel to side $BC$ of a triangle $ABC$, intersects $AB$ and $AC$ at $D$ and $E$ respectively. Prove that

State the negations for the following statements:

In each of the following questions, there are two statements. State if the second is the negation of the first or not.

Write the converses of the following statements.

Write the converses of the following statements. Also, decide in each case whether the converse is true or false.