# Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving. (a) For each positive real number x, if x is irrational, then x^2 is irrational. (b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational

Question
Irrational numbers
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then $$\displaystyle{x}^{{2}}$$ is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational

2020-12-15
(a) The given statement is false. Because for each positive real number x, if x is irrational number, then it is not necessary to their square $$\displaystyle{x}^{{2}}$$ is also irrational. For example:
consider $$\displaystyle{x}=\sqrt{{5}}$$ ((it is positive real number) Then,
$$\displaystyle{x}^{{2}}={\left(\sqrt{{5}}\right)}^{{2}}$$
$$\displaystyle{x}^{{2}}={5}$$ {since, $$\displaystyle{\left(\sqrt{{a}}\right)}^{{2}}={a}$$}
Since, 5 is not an irrational number because it can be written in the form of $$\displaystyle\frac{{p}}{{q}}$$ {where p and q are integer}.
That is $$\displaystyle{5}=\frac{{5}}{{1}}$$.
Hence, given statement is false.
(b)The given statement is true.To prove this, contrary assume that x and y both are rational number. So, $$\displaystyle{x}=\frac{{a}}{{b}}$$ {where a and b are the integers}
$$\displaystyle{y}=\frac{{c}}{{d}}$$ {where c and d are the integers}
According to the given statement:
x+y=irrational number
$$\displaystyle\frac{{a}}{{b}}+\frac{{c}}{{d}}$$=ZSK irrational number
$$\displaystyle\frac{{{a}{d}+{c}{d}}}{{{b}{d}}}=$$ irrational number
Then,
$$\displaystyle\frac{{p}}{{q}}=$$ irrational number
rational number=irrational number
Which is the contradiction. Therefore, our consideration values {x and y both are rational number} are wrong. Hence, the given statement is true.
For example:
$$\displaystyle{x}={2},{y}=\sqrt{{3}}$$, then $$\displaystyle{x}+{y}={2}+\sqrt{{3}}=$$ irrational

### Relevant Questions

True or False?
1) Let x and y real numbers. If $$\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}$$ and $$\displaystyle{x}\ne{y}$$, then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, $$\displaystyle{m}\in{Z}$$ such that m > 1. If $$\displaystyle{a}{c}\equiv{b}{c}{\left(\mod\right)},{t}{h}{e}{n}\ {a}\equiv{b}{\left(\mod{m}\right)}$$

The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance $$\displaystyle{R}_{{x}}$$ is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance $$\displaystyle{R}_{{y}}$$. The procedure for finding the unknown resistance $$\displaystyle{R}_{{x}}$$ is as follows. Measure resistance $$\displaystyle{R}_{{1}}$$ between points A and B. Then connect A and B with a heavy conducting wire and measure resistance $$\displaystyle{R}_{{2}}$$ between points A and C.Derive a formula for $$\displaystyle{R}_{{x}}$$ in terms of the observable resistances $$\displaystyle{R}_{{1}}$$ and $$\displaystyle{R}_{{2}}$$. A satisfactory ground resistance would be $$\displaystyle{R}_{{x}}{<}{2.0}$$ Ohms. Is the grounding of the station adequate if measurments give $$\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}$$ and R_2=6.0 Ohms?
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Which statement is false?
A. every irrational number is also a real number.
B. every integer is also a real number.
C. no irrational number is irrational.
D. every integer is also an irrational number.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
$$\int_{0}^{3}\sin(x^{2})dx=\int_{0}^{5}\sin(x^{2})dx+\int_{5}^{3}\sin(x^{2})dxZKS asked 2021-03-09 Discover prove: Combining Rational and Irrationalnumbers is \(\displaystyle{1.2}+\sqrt{{2}}$$ rational or irrational? Is $$\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}$$ rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.
Determine whether the below given statement is true or false. If the statement is false, make the necessary changes to produce a true statement:
All irrational numbers satisfy |x - 4| > 0.
The figure shows 3 crates being pushed over a concrete floor by a horizontal force f of magnitude 440N. The masses of the cratesare $$\displaystyle{m}_{{1}}={30}$$ kg, $$\displaystyle{m}_{{2}}={10}$$ kg, and $$\displaystyle{m}_{{3}}={20}$$ kg.The coefficient of kineticfriction between the floor and each of the crates is 0.7. a) what is the magnitude $$\displaystyle{F}_{{{32}}}$$ of the force on crate 3 from crate 2? b) If the crates then slide onto a polished floor, where the coefficientof kinetic friction is less than 0.700, is magnitude PSKF_{32}ZSL more than,less than, or the same as it was when the coeffient was 0.700?

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.