Download App

Appendix 2 — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAppendix 25 Marks
Question
Show that the set of all prime numbers is infinite.
✓

Answer

Let P be the set of all prime numbers. We take the negation of the statement “the set of all prime numbers is infinite”, i.e., we assume the set of all prime numbers to be finite. Hence, we can list all the prime numbers as $P_1 , P_2 , P_3 ,..., P_k$ (say). Note that we have assumed that there is no prime number other than $P_1 , P_2 , P_3 ,..., P_k$ . Now consider $N = (P_1 , P_2 , P_3 ,..., P_k + 1)$ ... (i)
N is not in the list as N is larger than any of the numbers in the list. N is either prime or composite. If N is a prime, then by (i), there exists a prime number which is not listed. On the other hand, if N is composite, it should have a prime divisor. But none of the numbers in the list can divide N, because they all leave the remainder 1. Hence, the prime divisor should be other than the one in the list. Thus, in both the cases whether N is a prime or a composite, we ended up with contradiction to the fact that we have listed all the prime numbers. Hence, our assumption that a set of all prime numbers is finite is false. Thus, the set of all prime numbers is infinite.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Appendix 2→Appendix 2 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{1}{1+\cot\text{x}}\text{ dx}$
→
Evaluate the following integrals:$\int\text{x}^2\sin^{-1}\text{x dx}$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{5\text{x}}{1+6\text{x}^3}\Big), -\frac{1}{\sqrt{6}}<\text{x}<\frac{1}{\sqrt{6}}$
→
If $\text{A}=\begin{bmatrix}3&-5\\-4&2\end{bmatrix},$ then find $A^2 - 5A - 14I$. Hence, obtain $A^3$.
→
$\int\frac{\text{x}-1}{\sqrt{\text{x}+4}}\ \text{dx}$
→
A particle moves along the curve $y = x^2 + 2x$. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?
→
If $\text{A}=\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix},\text{B}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $x^2 = -1$ then show that $(A + B)^2 = A^2 + B^2$.
→
Solve the differential equation $(1 + x^2) \frac{\text{dy}}{\text{dx}} + \text{y} = \text{e}^{\tan^{-1}\text{x}.}$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big)$
→
Evaluate :
$\int_0^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{3+\sin 2 x}\right) d x$
→