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 $$ \mathrm{Z} \cap \mathrm{N} $$ $$ 2.3 .4 \mathrm{~N}^{\prime} $$

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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,
     $$
     \mathrm{N}^{\prime} = \{ \dots, -3, -2, -1, 0 \}.
     $$
**Venn Diagram and Set Operations:** 1. **Venn Diagram:** - **Universe ($$ \mathrm{Z} $$):** Represents all integers. - **Whole Numbers ($$ \mathrm{N}_0 $$):** A subset of $$ \mathrm{Z} $$, including 0 and all positive integers. - **Natural Numbers ($$ \mathrm{N} $$):** A subset of $$ \mathrm{N}_0 $$, excluding 0. The diagram shows $$ \mathrm{N}_0 $$ inside $$ \mathrm{Z} $$ and $$ \mathrm{N} $$ inside $$ \mathrm{N}_0 $$. 2. **Set Operations:** - **2.3.1** $$ \mathrm{N} \cap \mathrm{N}_0 = \mathrm{N} $$ - **2.3.2** $$ \mathrm{Z} \cup \mathrm{N} = \mathrm{Z} $$ - **2.3.3** $$ \mathrm{Z} \cap \mathrm{N} = \mathrm{N} $$ - **2.3.4** $$ \mathrm{N}^{\prime} = \{ \dots, -3, -2, -1, 0 \} $$

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