If P=`{5,6,8,10,12,} , Q = {0,3,6,9,12,15,8} and R = { 0,1,2,3, ….10 }`
`n (P cap Q ) , n ( Q cap R) and n(P cap R)`

Answer

To solve the problem, we will find the intersections of the sets P, Q, and R step by step.

**Step 1: Identify the sets.**
- Set P = {5, 6, 8, 10, 12}
- Set Q = {0, 3, 6, 9, 12, 15, 8}
- Set R = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**Step 2: Find P ∩ Q (P intersection Q).**
- We need to find the common elements between sets P and Q.
- The elements in P are {5, 6, 8, 10, 12} and in Q are {0, 3, 6, 9, 12, 15, 8}.
- The common elements are 6, 8, and 12.
- Therefore, P ∩ Q = {6, 8, 12}.

**Step 3: Count the number of elements in P ∩ Q.**
- The number of elements in P ∩ Q is 3.
- So, n(P ∩ Q) = 3.

**Step 4: Find Q ∩ R (Q intersection R).**
- We need to find the common elements between sets Q and R.
- The elements in Q are {0, 3, 6, 9, 12, 15, 8} and in R are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
- The common elements are 0, 3, 6, 8, and 9.
- Therefore, Q ∩ R = {0, 3, 6, 8, 9}.

**Step 5: Count the number of elements in Q ∩ R.**
- The number of elements in Q ∩ R is 5.
- So, n(Q ∩ R) = 5.

**Step 6: Find P ∩ R (P intersection R).**
- We need to find the common elements between sets P and R.
- The elements in P are {5, 6, 8, 10, 12} and in R are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
- The common elements are 5, 6, 8, and 10.
- Therefore, P ∩ R = {5, 6, 8, 10}.

**Step 7: Count the number of elements in P ∩ R.**
- The number of elements in P ∩ R is 4.
- So, n(P ∩ R) = 4.

**Final Answers:**
- n(P ∩ Q) = 3
- n(Q ∩ R) = 5
- n(P ∩ R) = 4

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