20. Suppose \( \xi=\{x: x \) is an integer such that \( 0

Ask by Spencer Bowers. in Singapore

Mar 20,2025

**(a)** - For set \( A \): We have \[ 5x < 38 \quad \Longrightarrow \quad x < \frac{38}{5} = 7.6. \] Since \( x \) is an integer and \( 0 < x \leq 10 \), the elements of \( A \) are \[ A = \{1, 2, 3, 4, 5, 6, 7\}. \] - For set \( B \): The condition is \[ 2 < x < 7. \] Thus the integer values of \( x \) satisfying this inequality are \[ B = \{3, 4, 5, 6\}. \] **(b)** Since every element of \( B \) (i.e. \( 3, 4, 5, 6 \)) is also in \( A \) (because \( A \) contains \( 1, 2, 3, 4, 5, 6, 7 \)), we deduce that \[ B \subset A. \] **(c)** To illustrate this with a Venn diagram, consider the universal set \[ \xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. \] - Draw a rectangle to represent \( \xi \). - Inside the rectangle, draw a circle labeled \( A \) containing the elements \( \{1, 2, 3, 4, 5, 6, 7\} \). - Within the circle \( A \), draw a smaller circle labeled \( B \) containing the elements \( \{3, 4, 5, 6\} \). A simplified diagram can be represented as follows: ``` -------------------------- | ξ | | 8, 9, 10 | | __________ | | | A | | | | 1,2,3,4,5,6,7 | | | ______ | | | | B | | | | |3,4,5,6| | | | |______| | | | | -------------------------- ``` Here, the circle \( B \) is completely inside the circle \( A \), illustrating that \( B \subset A \) within the universal set \( \xi \).