Animated Solution for Mathematics - Sets and Relations: Let $A=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}:|x|+|y|=3\}$ and $B=\{(x,y)\in \mathbb{Z}\times \mathbb{Z}:|x|+|y|\le 3\}$. If $C=\{(x,y)\in A\cap B:x=0\text{ or }y=0\}$, then ${\sum }_{(x,y)\in C}|x+y|$ is :

We are dealing with sets of points in the integer grid $\mathbb{Z}\times \mathbb{Z}$. The equation $|x|+|y|=3$ defines a shape known as a diamond, which is a square rotated by $45^{\circ }$.

Imagine standing at the origin $(0,0)$ and walking exactly three units in any combination of directions—up, down, left, or right. The boundary of this movement is our Set $A$. It is a discrete collection of points, as we are restricted to integer coordinates.

Next, we encounter Set $B$, defined by the inequality $|x|+|y|\le 3$. If Set $A$ is the fence around our diamond, Set $B$ is the entire field inside the fence, including the fence itself.

When we look for the intersection $A\cap B$, we are identifying points that exist in both the boundary and the region. Since the boundary is already contained within the region, the intersection is simply the boundary itself:

The interior points of $B$ are irrelevant here because they do not satisfy the equality condition required by Set $A$.

Finding Set $C$

We define Set $C$ as the subset of $A\cap B$ where $x=0$ or $y=0$. Geometrically, we are looking for the points on our diamond that lie exactly on the coordinate axes.

If $x=0$, our boundary equation $|x|+|y|=3$ simplifies to: This yields two points: $(0,3)$ and $(0,-3)$.

If $y=0$, the equation becomes: This yields two additional points: $(3,0)$ and $(-3,0)$.

Thus, Set $C$ contains exactly four points: $\{(3,0),(-3,0),(0,3),(0,-3)\}$.

We must calculate the sum ${\sum }_{(x,y)\in C}|x+y|$. Let us evaluate this for each point in Set $C$:

1. For $(3,0)$: $|3+0|=3$ 2. For $(-3,0)$: $|-3+0|=|-3|=3$ 3. For $(0,3)$: $|0+3|=3$ 4. For $(0,-3)$: $|0-3|=|-3|=3$

Due to the inherent symmetry of the diamond, every point in our set yields a value of $3$. To find the total sum, we add these values:

The final result is 12.