Let A = {2, 4, 5, 6}, B = {1, 3, 5, 7}, C = {3, 4, 6, 9, 10} and D = {1, 2, 3, 7, 8, 9}. Find A ∪ B, B ∪ C, C ∪ D, A ∩ B, C ∩ D and D ∩ A.

Step-by-step solution

1. 1

   Fix the two definitions before computing. $A\cup B$ ("union") contains every element that is in $A$ or $B$ or both, each listed once. $A\cap B$ ("intersection") contains only the elements in both. A reliable habit is to write the answers in increasing order so that a missing or repeated element is easy to spot.

2. 2

   Compute the three unions. Merge the lists and delete repeats:

   $A\cup B=\{1,2,3,4,5,6,7\}$

   $B\cup C=\{1,3,4,5,6,7,9,10\}$

   $C\cup D=\{1,2,3,4,6,7,8,9,10\}$

   Note $5$ appears in both $A$ and $B$ but is written only once in $A\cup B$.

3. 3

   Compute A ∩ B. Scan $A=\{2,4,5,6\}$ against $B=\{1,3,5,7\}$: only $5$ occurs in both.

   $A\cap B=\{5\}$

4. 4

   Compute C ∩ D. $C=\{3,4,6,9,10\}$ and $D=\{1,2,3,7,8,9\}$ share $3$ and $9$.

   $C\cap D=\{3,9\}$

5. 5

   Compute D ∩ A. $D=\{1,2,3,7,8,9\}$ and $A=\{2,4,5,6\}$ share only $2$. Intersection is commutative, so $D\cap A=A\cap D$.

   $D\cap A=\{2\}$

   $\boxed{A\cup B=\{1,\ldots,7\},\ B\cup C=\{1,3,4,5,6,7,9,10\},\ C\cup D=\{1,2,3,4,6,7,8,9,10\},\ A\cap B=\{5\},\ C\cap D=\{3,9\},\ D\cap A=\{2\}}$

6. 6

   Check each pair with the counting identity. $|X\cup Y|=|X|+|Y|-|X\cap Y|$. For $A$ and $B$: $4+4-1=7$ ✓. For $C$ and $D$: $5+6-2=9$ ✓. For $B$ and $C$: $|B\cap C|=|\{3\}|=1$, so $4+5-1=8$ ✓. All three unions have the predicted sizes.