(a) How would you show that, for any whole number $$ a, a+0=a $$ if you were starting from our definition of addition and the audience is your university class? (b) How would you show that, for any whole number $$ a, a+0=a $$ if you were talking to your class of fourth graders? (a) Choose the correct answer below. A. Use manipulatives to show that the union of a collection with a collection with no elements has the same number of elements as the first collection. B. 0 is the cardinal number of the empty set: that is, $$ n(\varnothing)=0 $$. If the set $$ A $$ has a elements in it $$ (n(A)=a) $$, then $$ n(A)+n(\varnothing)=n(A \cup \varnothing) $$, because $$ A $$ and $$ \varnothing $$ are disjoint. But $$ A \cup \varnothing=A $$, since there are no elements in $$ \varnothing $$ to add to the set $$ A $$. Thus, $$ n(A \cup \varnothing)=n(A)=a $$. C. Start with two disjoint sets of the same cardinality $$ A $$ and $$ B $$ with $$ n(A)=n(B)=a $$. Then write the equation. Since $$ n(A)=n(B)=a, n(A \cup B)=a $$. Since they are disjoint, $$ A \cap B=\varnothing $$. Then, $$ a+0=n(A \cup(A \cap B)) $$ which is equal to $$ n(A \cup B)=a $$. D. 0 is the cardinal number of the empty set: that is, $$ n(\varnothing)=0 $$. If the set $$ A $$ has a elements in it $$ (n(A)=a) $$, then $$ n(A)+n(\varnothing)=n(A \cap \varnothing) $$, because $$ A $$ and $$ \varnothing $$ are disjoint. But $$ A \cap \varnothing=A $$, since there are no elements in $$ \varnothing $$ to add to the set $$ A $$. Thus, $$ n(A \cup \varnothing)=n(A)=a $$.

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To address the problem, we will break it down into two parts: (a) for a university class and (b) for a class of fourth graders.

### (a) University Class Explanation

To show that for any whole number $$ a $$, $$ a + 0 = a $$, we can use the concept of cardinality of sets. The most appropriate answer from the options provided is:

**B.** 0 is the cardinal number of the empty set: that is, $$ n(\varnothing)=0 $$. If the set $$ A $$ has $$ a $$ elements in it $$ (n(A)=a) $$, then $$ n(A)+n(\varnothing)=n(A \cup \varnothing) $$, because $$ A $$ and $$ \varnothing $$ are disjoint. But $$ A \cup \varnothing=A $$, since there are no elements in $$ \varnothing $$ to add to the set $$ A $$. Thus, $$ n(A \cup \varnothing)=n(A)=a $$.

**Explanation:**
1. Define the empty set $$ \varnothing $$ and its cardinality $$ n(\varnothing) = 0 $$.
2. Let $$ A $$ be a set with $$ a $$ elements, so $$ n(A) = a $$.
3. Since $$ A $$ and $$ \varnothing $$ are disjoint, the union of these sets is simply $$ A $$ itself: $$ A \cup \varnothing = A $$.
4. Therefore, the cardinality of the union is $$ n(A \cup \varnothing) = n(A) = a $$.
5. This leads to the equation $$ n(A) + n(\varnothing) = n(A) $$, which simplifies to $$ a + 0 = a $$.

### (b) Fourth Graders Explanation

To explain $$ a + 0 = a $$ to fourth graders, we can use a more visual and intuitive approach:

1. **Use Manipulatives:** Gather some objects (like blocks or counters) to represent the number $$ a $$. For example, if $$ a = 3 $$, show three blocks.
2. **Introduce Zero:** Now, show them an empty space or an empty box to represent zero. Explain that zero means "nothing" or "no blocks."
3. **Combine the Sets:** Ask the students to combine the blocks with the empty box. Since there are no blocks in the empty box, the total number of blocks remains the same.
4. **Conclusion:** Reinforce that adding zero (the empty box) to the three blocks still gives them three blocks. Thus, $$ 3 + 0 = 3 $$.

This hands-on approach helps them visualize the concept of adding zero and reinforces the idea that adding nothing to a number does not change its value.
(a) For a university class, the correct answer is: **B.** 0 is the cardinal number of the empty set: that is, $$ n(\varnothing)=0 $$. If the set $$ A $$ has $$ a $$ elements in it $$ (n(A)=a) $$, then $$ n(A)+n(\varnothing)=n(A \cup \varnothing) $$, because $$ A $$ and $$ \varnothing $$ are disjoint. But $$ A \cup \varnothing=A $$, since there are no elements in $$ \varnothing $$ to add to the set $$ A $$. Thus, $$ n(A \cup \varnothing)=n(A)=a $$. (b) For a fourth-grade class, explain that adding zero to any number doesn't change its value. For example, if you have 3 blocks and add zero blocks, you still have 3 blocks. So, $$ 3 + 0 = 3 $$.

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