miniliv4

2022-09-04

Is there a natural number between 0 and 1?

A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)

A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)

graulhavav9

Beginner2022-09-05Added 14 answers

Every natural number $m$ is either 0 or $s(n)$, where n is a natural number.

Proof: It can't be both, because $s(n)$ can't be 0. Set of all natural numbers which are either 0 or $s(n)$ for some $n$ satisfies induction principle, so it contains all natural numbers.

Direct consequence: Every natural number is either 0, or $s(0)$ or $s(s(n))$ for some natural number $n$.

Suppose there is $m$ such that $0<m<s(0)$. Either $m$ is 0, $s(0)$ or $s(s(n))$. First two cannot hold, so you have $s(s(n))<s(0)$, i.e., $s(n)<0$.

Proof: It can't be both, because $s(n)$ can't be 0. Set of all natural numbers which are either 0 or $s(n)$ for some $n$ satisfies induction principle, so it contains all natural numbers.

Direct consequence: Every natural number is either 0, or $s(0)$ or $s(s(n))$ for some natural number $n$.

Suppose there is $m$ such that $0<m<s(0)$. Either $m$ is 0, $s(0)$ or $s(s(n))$. First two cannot hold, so you have $s(s(n))<s(0)$, i.e., $s(n)<0$.

s2vunov

Beginner2022-09-06Added 2 answers

HINT

$\text{}\text{}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}\mathrm{n}\text{}=\text{}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}0\text{}\Rightarrow \text{}\mathrm{n}\phantom{\rule{mediummathspace}{0ex}}=\phantom{\rule{mediummathspace}{0ex}}0\phantom{\rule{mediummathspace}{0ex}}\ne \phantom{\rule{mediummathspace}{0ex}}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}\mathrm{m}$

$\text{}\text{}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}\mathrm{n}\text{}=\text{}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}0\text{}\Rightarrow \text{}\mathrm{n}\phantom{\rule{mediummathspace}{0ex}}=\phantom{\rule{mediummathspace}{0ex}}0\phantom{\rule{mediummathspace}{0ex}}\ne \phantom{\rule{mediummathspace}{0ex}}\mathrm{S}\phantom{\rule{mediummathspace}{0ex}}\mathrm{m}$

The transverse axis of a hyperbola is double the conjugate axes. Whats the eccentricity of the hyperbola $A)\frac{\sqrt{5}}{4}\phantom{\rule{0ex}{0ex}}B)\frac{\sqrt{7}}{4}\phantom{\rule{0ex}{0ex}}C)\frac{7}{4}\phantom{\rule{0ex}{0ex}}D)\frac{5}{3}$

Which of the following is an example of a pair of parallel lines?

Sides of a triangle

Surface of a ball

Corner of a room

Railway trackState whether the statements are True or False. All rectangles are parallelograms.

Point T(3, −8) is reflected across the x-axis. Which statements about T' are true?

If two parallel lines are cut by a transversal, then:

each pair of alternate angles are equal

each pair of alternate angles add up to 180 degrees

each pair of corresponding angles add up to 180 degees.

each pair of corresponding angles is equal.The opposite faces of a dice always have a total of ___ on them.

What is the term for a part of a line with two endpoints?

A. Line Segment

B. Line

C. Ray

D. PointABCD is a parallelogram. Which of the following is not true about ABCD?

AB=CD

AD=CB

AD||CD

AB||CDIf the points $(a,0)$, $(0,b)$ and $(1,1)$ are collinear then which of the following is true?

$\frac{1}{a}+\frac{1}{b}=1$

$\frac{1}{a}-\frac{1}{b}=2$

$\frac{1}{a}-\frac{1}{b}=-1$

$\frac{1}{a}+\frac{1}{b}=2$The angles of a triangle are in the ratio 1 : 1 : 2. What is the largest angle in the triangle?

$A){90}^{\circ}\phantom{\rule{0ex}{0ex}}B){135}^{\circ}\phantom{\rule{0ex}{0ex}}C){45}^{\circ}\phantom{\rule{0ex}{0ex}}D){150}^{\circ}$What is the greatest number of acute angles that a triangle can contain?

How many right angles are required to create a complete angle?

1) Two

2) Three

3) Four

4) FiveFind the sum of interior angles of a polygon with 12 sides in degrees.${1800}^{\circ}\phantom{\rule{0ex}{0ex}}{1200}^{\circ}\phantom{\rule{0ex}{0ex}}{1500}^{\circ}\phantom{\rule{0ex}{0ex}}{1400}^{\circ}$

In Euclid's Division Lemma when $a=bq+r$ where $a,b$ are positive integers then what values $r$ can take?

When two lines cross each other at a point we call them ___ lines.