Q4(ii):

State whether the following statements are true or false. Give reasons for your answers. (ii) Every integer is a whole number.

Step 1: Formal Definitions of the Number Sets

To evaluate the validity of the statement, we must first establish the rigorous mathematical definitions of the two sets of numbers in question: Integers and Whole Numbers.

• Integers ($\mathbb{Z}$): The set of integers includes all positive counting numbers, their negative counterparts, and zero. It extends infinitely in both positive and negative directions.
  $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
• Whole Numbers ($\mathbb{W}$): The set of whole numbers includes all natural (counting) numbers and zero. It does not contain any negative numbers.
  $\mathbb{W} = \{0, 1, 2, 3, \dots\}$

Step 2: Set-Theoretic Analysis

The statement "Every integer is a whole number" can be translated into set theory as a subset proposition: $\mathbb{Z} \subseteq \mathbb{W}$ [Read: The set of Integers is a subset of the set of Whole Numbers].

For $\mathbb{Z} \subseteq \mathbb{W}$ to be true, every element $x$ that belongs to $\mathbb{Z}$ must also belong to $\mathbb{W}$ ($\forall x \in \mathbb{Z} \implies x \in \mathbb{W}$).

By comparing the sets defined in Step 1, we observe the following relationship:

• All elements of $\mathbb{W}$ exist within $\mathbb{Z}$. Therefore, $\mathbb{W} \subset \mathbb{Z}$ is a true statement [Every whole number is an integer].
• However, $\mathbb{Z}$ contains negative numbers (e.g., $-1, -2, -3$) which do not exist in $\mathbb{W}$. Therefore, $\mathbb{Z} \not\subseteq \mathbb{W}$.

Step 3: Visual Proof via Number Line

The following high-precision number line illustrates the spatial and logical boundaries of both sets. Notice that the domain of Integers extends into the negative axis, whereas Whole Numbers are strictly bounded at zero and extend only in the positive direction.

Step 4: Counterexample Formulation

In formal logic, a universal affirmative statement ("Every A is B") is proven false if even a single counterexample can be provided. We must find an integer that is not a whole number.

Let $x = -5$.

• Is $-5 \in \mathbb{Z}$? Yes, $-5$ is a negative integer.
• Is $-5 \in \mathbb{W}$? No, the set of whole numbers $\mathbb{W}$ does not contain negative values.

Because we have identified an element that belongs to $\mathbb{Z}$ but not to $\mathbb{W}$, the statement "Every integer is a whole number" is logically invalid.

Final Solution: False. The statement is false because the set of integers includes negative numbers (e.g., $-1, -2, -3$), whereas the set of whole numbers only includes zero and positive counting numbers. Therefore, negative integers are not whole numbers.