(b) Let the universal set \( \mathcal{U} \) be the set of real numbers and let \( A=\{x \in \mathbb{R} \) : \( x>0\} \). Find \( A^{c} \). (c) Show the intersection of \( A, B \) and \( C \) in venn diagram.

Let's solve the problems step by step. ### Part (b) We are given the universal set \( \mathcal{U} \) as the set of real numbers and the set \( A \) defined as: \[ A = \{ x \in \mathbb{R} : x > 0 \} \] To find the complement of set \( A \), denoted as \( A^c \), we need to identify all elements in the universal set \( \mathcal{U} \) that are not in \( A \). Since \( A \) consists of all positive real numbers, its complement \( A^c \) will include: 1. All negative real numbers. 2. The number zero. Thus, we can express \( A^c \) as: \[ A^c = \{ x \in \mathbb{R} : x \leq 0 \} \] ### Part (c) To show the intersection of sets \( A \), \( B \), and \( C \) in a Venn diagram, we need to define the sets \( B \) and \( C \). However, since the definitions of \( B \) and \( C \) are not provided, I will describe how to represent the intersection in a Venn diagram. 1. **Draw three overlapping circles**: Each circle represents one of the sets \( A \), \( B \), and \( C \). 2. **Label the circles**: Label them as \( A \), \( B \), and \( C \). 3. **Identify the intersection**: The area where all three circles overlap represents the intersection \( A \cap B \cap C \). If you provide the definitions of sets \( B \) and \( C \), I can give a more specific representation of the intersection. Would you like to provide those definitions or need further assistance?