Related questions with answers

Refer to the Early Childhood Education Journal (Mar. 2014) study of interactions in a children's museum. (Interactions by visitors to the museum included show-and-tell, learning, teaching, refocusing, participatory play, advocating, and disciplining interactions.) Recall that the researchers observed a sample of 170 meaningful interactions, of which 81 were led by children and 89 were led by adult caregivers. a. Give a point estimate of the true proportion of all meaningful interactions in a children's museum that are led by children. b. Form a 90% confidence interval around the proportion, part a. c. Interpret the interval, part b, in the words of the study. d. Suppose an educator claims that the true proportion of all meaningful interactions in a children's museum that are led by children is 35%. Make an inference about this claim.

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Prove that if r and s are relatively prime, then $\varphi(r s)=\varphi(r) \varphi(s).$

Question

If a = bc with $a \neq 0$ and b and c nonunits, show that a is not an associate of b.

Solution

Verified

Step 1

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$\textbf{Given}$ : Let $R$ is an integral domain such that $\>a=bc\>$ for some elements $a \neq 0$ and $b,c$ are non-units in $R$ .

$\textbf{To Prove}$ : $a$ is not an associate of $b$ .

$\textbf{Proof}$ : If possible, let us assume that $a$ is an associate of $b$ in $R$ . Then there exists an unit $u$ in $R$ such that

a=bu.

Now by the given condition note that

\begin{align*} a=bc \>\> \text{ and }\> a=bu \implies \> & bc=bu \\ \implies \> & b(c-u)=0 \\ \implies \> & c=u,\>\> \text{ since $R$ is an integral domain and $a \neq 0$ in }R.\end{align*}

This contradicts the fact that $c$ is a non unit in $R$ . So our assumption that, $a$ is an associate of $b$ in $R$ is wrong. Hence, $a$ is not an associate of $b$ in $R$ . This completes the solution.