Example Question - roster form
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Set Theory Problem Involving Intersection of Sets in Roster Form
<p>To find \( A \cap B \) in roster form, identify the elements common to both set \( A \) and set \( B \).</p> <p>By definition from the question:</p> <p>\( A = \{x: x \leq 10, x \in \mathbb{N}, x \text{ is a prime number}\} \)</p> <p>\( B = \{x: x \in \mathbb{N}, x \text{ is even}\} \)</p> <p>First list the prime numbers less than or equal to 10 (prime numbers are numbers greater than 1 that have no positive divisors other than 1 and itself): 2, 3, 5, 7</p> <p>Now list the even natural numbers (since only 2 is both even and prime): 2</p> <p>Therefore, the intersection of set A and set B, \( A \cap B \), which includes all the elements that are both prime and even is: {2}, since 2 is the only even prime number.</p> <p>\( A \cap B = \{2\} \).</p>
Solving Set Theory Problems with Roster Notation
<p>\text{There are five problems to solve in this question.}</p> <p>\text{1. Convert the set-builder notation } \{x | x \text{ is a natural number less than 5}\} \text{ to roster form. }</p> <p>\text{Solution: } \{1, 2, 3, 4\}</p> <p>\text{2. Express the set } G = \{x | x \text{ is an even number between 1 and 20}\} \text{ in roster form. }</p> <p>\text{Solution: } \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}</p> <p>\text{3. Express the set } L = \{x | x \text{ is a letter in the word "MATHEMATICS"}\} \text{ in roster form. }</p> <p>\text{Solution: } \{'M', 'A', 'T', 'H', 'E', 'I', 'C', 'S'\}</p> <p>\text{4. Convert the set-builder notation } \{x | x \text{ is an odd integer greater than 5 and less than 15}\} \text{ to roster form. }</p> <p>\text{Solution: } \{7, 9, 11, 13\}</p> <p>\text{5. If } A = \{x | x \text{ is a prime number less than 10}\}, \text{ what is } A? </p> <p>\text{Solution: } \{2, 3, 5, 7\}</p>
Writing Sets of Multiples in Roster and Set-Builder Notations
<p>To find all multiples of \(3\) between \(1\) and \(20\), we list the multiples within this range.</p> <p>Roster form: \(\{3, 6, 9, 12, 15, 18\}\)</p> <p>To express this set in set-builder notation, we define the properties that all elements of the set share. All elements \(x\) are multiples of \(3\), hence \(x = 3n\) for some integer \(n\), and \(x\) is greater than \(1\) and less than or equal to \(20\).</p> <p>Set-builder form: \(\{x \mid x = 3n, n \in \mathbb{Z}, 1 < x \leq 20\}\)</p>
Conversion between Roster Form and Set-Builder Notation
<p>\begin{align*} A &= \{1, 3, 5, 7, 9\} \text{ in set-builder notation is } A = \{ x | x \text{ is an odd number less than 10} \}. \end{align*}</p> <p>\begin{align*} B &= \{a, e, i, o, u\} \text{ in set-builder notation is } B = \{ x | x \text{ is a vowel in the English alphabet} \}. \end{align*}</p> <p>\begin{align*} C &= \{2, 4, 6, 8, 10\} \text{ in set-builder notation is } C = \{ x | x \text{ is an even number less than or equal to 10} \}. \end{align*}</p> <p>\begin{align*} D &= \{x | x \text{ is a prime number less than 10}\} \text{ in roster form is } D = \{2, 3, 5, 7\}. \end{align*}</p> <p>\begin{align*} E &= \{10, 20, 30, 40, 50\} \text{ in set-builder notation is } E = \{ x | x = 10n, n \in \mathbb{N}, 1 \leq n \leq 5 \}. \end{align*}</p>
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