1. **Problem Statement:** Find the intersection $A \cap B$ of the sets $A$ and $B$ given as: $$A = \{a:\{1,2,3,4,5\}, b:\{1,2,3\}, c:\{0,1,2,3\}, d:\{3\}\}$$ $$B = \{a:\{1,2,3,4\}, b:\{1,2,3\}, c:\{0,1,2,3\}, d:\{0\}\}$$ 2. **Formula and Rules:** The intersection of two sets $X$ and $Y$, denoted $X \cap Y$, is the set of elements that are in both $X$ and $Y$. For each element label (a, b, c, d), we find the intersection of the corresponding subsets. 3. **Step-by-step Calculation:** - For $a$: $\{1,2,3,4,5\} \cap \{1,2,3,4\} = \{1,2,3,4\}$ - For $b$: $\{1,2,3\} \cap \{1,2,3\} = \{1,2,3\}$ - For $c$: $\{0,1,2,3\} \cap \{0,1,2,3\} = \{0,1,2,3\}$ - For $d$: $\{3\} \cap \{0\} = \emptyset$ 4. **Final Answer:** $$A \cap B = \{a:\{1,2,3,4\}, b:\{1,2,3\}, c:\{0,1,2,3\}, d:\emptyset\}$$ This means the intersection contains all elements common to both sets for each label, with $d$ having no common elements.