Sketch a Venn diagram of the following: natural numbers in one collection, whole numbers in another collection and integers as the universal collection. Use the following notation: Natural numbers \( =\mathrm{N} \) Whole numbers \( =\mathrm{N}_{0} \) Integers = Z Determine 2.3.1 \( \mathrm{N} \cap \mathrm{N}_{0} \) 2.3.2 ZUN 2.3.3 \( \mathrm{Z} \cap \mathrm{N} \) \( 2.3 .4 \mathrm{~N}^{\prime} \)

1. **Venn Diagram Description:** - The universe is the set of all integers, \( \mathrm{Z} \). - The set of whole numbers, \( \mathrm{N}_{0} \), is a subset of \( \mathrm{Z} \). - The set of natural numbers, \( \mathrm{N} \), is a proper subset of \( \mathrm{N}_{0} \) (since typically \( \mathrm{N} = \{1, 2, 3, \dots\} \) and \( \mathrm{N}_0 = \{0, 1, 2, 3, \dots\} \)). The diagram can be visualized as three concentric figures: - The outermost rectangle represents \( \mathrm{Z} \). - A circle inside represents \( \mathrm{N}_0 \). - A smaller circle inside \( \mathrm{N}_0 \) represents \( \mathrm{N} \). 2. **Set Operations:** **2.3.1** \(\mathrm{N} \cap \mathrm{N}_{0}\) - Since every element of \(\mathrm{N}\) is also an element of \(\mathrm{N}_0\), we have \[ \mathrm{N} \cap \mathrm{N}_{0} = \mathrm{N}. \] **2.3.2** \( \mathrm{Z} \cup \mathrm{N}\) - Since \( \mathrm{N} \subset \mathrm{Z} \), the union is just \( \mathrm{Z} \): \[ \mathrm{Z} \cup \mathrm{N} = \mathrm{Z}. \] **2.3.3** \(\mathrm{Z} \cap \mathrm{N}\) - The intersection of \( \mathrm{Z} \) with \( \mathrm{N} \) is simply the set \( \mathrm{N} \) itself: \[ \mathrm{Z} \cap \mathrm{N} = \mathrm{N}. \] **2.3.4** \( \mathrm{N}^{\prime} \) - Here, \( \mathrm{N}^{\prime} \) denotes the complement of \( \mathrm{N} \) relative to the universal set \( \mathrm{Z} \). That is, \[ \mathrm{N}^{\prime} = \mathrm{Z} \setminus \mathrm{N}. \] - Given \( \mathrm{N} = \{1, 2, 3, \dots\} \), the complement in \( \mathrm{Z} \) consists of all integers that are not natural numbers. Therefore, \[ \mathrm{N}^{\prime} = \{ \dots, -3, -2, -1, 0 \}. \]