Pregunta
\begin{tabular}{l} Question \\ Which of the following correctly describes the set \( \{2 n \mid n \in \mathbb{Z}\} \) ? \\ Select the correct answer below: \\ The set of all odd integers \\ The set of all integers greater than 2 . \\ The set of all real numbers greater than 2. \\ The set of all even integers. \\ \hline\end{tabular}
Ask by Garza Ayala. in the United States
Mar 09,2025
Solución de inteligencia artificial de Upstudy
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The set consists of all even integers.
Solución
1. The set is defined as \( \{2 n \mid n \in \mathbb{Z}\} \), which means we take every integer \( n \) from the set of integers \( \mathbb{Z} \) and multiply it by 2.
2. Multiplying any integer by 2 always gives an even number. For example, if \( n = 1 \), then \( 2(1) = 2 \) (even); if \( n = -3 \), then \( 2(-3) = -6 \) (even).
3. Therefore, all numbers in the set are even.
4. Comparing with the answer choices, the option that correctly describes this set is:
\( \boxed{\text{The set of all even integers.}} \)
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Simplifique esta solución Beyond the Answer
The set \( \{2 n \mid n \in \mathbb{Z}\} \) is the set of all even integers. This is because for any integer \( n \), multiplying by 2 results in an even number. Thus, whether \( n \) is positive, negative, or zero, the output will always be an even integer like -4, 0, or 10. If you're learning about set notation, remember that understanding the operations involved helps clarify what each set represents. For example, knowing that \( \mathbb{Z} \) represents all integers and that multiplying by 2 generates even results can be a handy shortcut for identifying similar expressions!
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