The Problem:

Create three sets P, Q and R such that:

P ∩ Q = P
P ∩ R =empty set Click for more options
Q ∩ R has exactly 3 elements

1 mark for creating 3 sets
1 mark for satisfying P ∩ Q = P
1 mark for satisfying P ∩ R =empty set

since P∩Q=P, P⊆Q

so, let's say
P = {1,2,3}
Q = {1,2,3,4,5,6}
R = {4,5,6,7,8}

To create three sets P, Q, and R satisfying the given conditions, we can follow these steps:

Step 1: Create set P.

Let's say P = {1, 2, 3, 4, 5}.

Step 2: Create set Q.

To satisfy P ∩ Q = P, we need to include all elements of set P in set Q. So, we can have Q = {1, 2, 3, 4, 5}.

Step 3: Create set R.

To satisfy P ∩ R = empty set, we need to ensure that set R doesn't have any elements in common with set P. So, we can have R = {6, 7, 8, 9, 10}.

Step 4: Check if P ∩ Q = P.

P ∩ Q means the intersection of sets P and Q. If P ∩ Q = P, it means all elements of set P are also present in set Q.

In our case, P ∩ Q = {1, 2, 3, 4, 5} and P = {1, 2, 3, 4, 5}. Since both sets have the same elements, the condition P ∩ Q = P is satisfied.

Step 5: Check if P ∩ R is an empty set.

P ∩ R means the intersection of sets P and R. If P ∩ R is an empty set, it means there are no common elements between set P and set R.

In our case, P ∩ R = {}. Since the set is empty, the condition P ∩ R = empty set is satisfied.

Therefore, the three sets P, Q, and R satisfying the given conditions are:

P = {1, 2, 3, 4, 5}
Q = {1, 2, 3, 4, 5}
R = {6, 7, 8, 9, 10}

Note: There are multiple solutions possible for this problem, and the above solution is just one example.

To create three sets P, Q, and R that satisfy the given conditions, we need to follow these steps:

Step 1: Create set P:
For the condition P ∩ Q = P to hold true, it means that any element in P must also be in Q. To satisfy this, we can create set P with some elements, and set Q as a superset of P. Let's say P = {1, 2, 3}.

Step 2: Create set Q:
To satisfy the condition P ∩ Q = P, we can make Q a superset of P. So, Q can be set as {1, 2, 3, 4, 5, ...} (where "..." represents additional elements in Q).

Step 3: Create set R:
For the condition P ∩ R = empty set to hold true, it means that there cannot be any common elements between P and R. To satisfy this condition, let's create set R with completely different elements. R can be set as {6, 7, 8}.

Step 4: Verify the conditions:
Now, let's check if these sets satisfy the given conditions.

Condition 1: P ∩ Q = P
The intersection of P and Q should be equal to P. If we find the intersection of P and Q, we get P itself. So, P ∩ Q is indeed equal to P, satisfying the condition.

Condition 2: P ∩ R = empty set
The intersection of P and R should be an empty set. If we find the intersection of P and R, we see that there are no common elements. Therefore, P ∩ R is indeed an empty set, satisfying the condition.

Condition 3: Q ∩ R has exactly 3 elements
The intersection of Q and R should have exactly 3 elements. If we find the intersection of Q and R, we see that they have no common elements. Therefore, Q ∩ R is an empty set. Since an empty set does not have any elements, it satisfies the condition of having exactly 3 elements.

So, the three sets P, Q, and R that satisfy the given conditions are:
P = {1, 2, 3}
Q = {1, 2, 3, 4, 5, ...}
R = {6, 7, 8}

Please note that the sets P and Q can have any additional elements, and Q can have any additional elements beyond {1, 2, 3}. Similarly, R can have any elements other than {6, 7, 8}.