Let A = { x: x is a multiple of 2, x < 15 } ,
B = {x:x is a multiple of 3,x < 20 },
C = { x : x is a prime x < 20 },
Find
(a) ` A uu B (b) A uu C ( c ) B uu C (d) A nn B (e ) A nn C (f) B nn C `

Answer

To solve the problem, we will first define the sets A, B, and C in roster form, and then we will find the required unions and intersections step by step.

### Step 1: Define the sets in roster form

1. **Set A**: A = { x: x is a multiple of 2, x < 15 }
- The multiples of 2 less than 15 are: 2, 4, 6, 8, 10, 12, 14.
- So, A = {2, 4, 6, 8, 10, 12, 14}.

2. **Set B**: B = { x: x is a multiple of 3, x < 20 }
- The multiples of 3 less than 20 are: 3, 6, 9, 12, 15, 18.
- So, B = {3, 6, 9, 12, 15, 18}.

3. **Set C**: C = { x: x is a prime number, x < 20 }
- The prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19.
- So, C = {2, 3, 5, 7, 11, 13, 17, 19}.

### Step 2: Find the required unions and intersections

(a) **A ∪ B** (Union of A and B):
- Combine all elements from both sets A and B, removing duplicates.
- A ∪ B = {2, 4, 6, 8, 10, 12, 14} ∪ {3, 6, 9, 12, 15, 18} = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18}.

(b) **A ∪ C** (Union of A and C):
- Combine all elements from both sets A and C, removing duplicates.
- A ∪ C = {2, 4, 6, 8, 10, 12, 14} ∪ {2, 3, 5, 7, 11, 13, 17, 19} = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 19}.

(c) **B ∪ C** (Union of B and C):
- Combine all elements from both sets B and C, removing duplicates.
- B ∪ C = {3, 6, 9, 12, 15, 18} ∪ {2, 3, 5, 7, 11, 13, 17, 19} = {2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}.

(d) **A ∩ B** (Intersection of A and B):
- Find common elements in sets A and B.
- A ∩ B = {2, 4, 6, 8, 10, 12, 14} ∩ {3, 6, 9, 12, 15, 18} = {6, 12}.

(e) **A ∩ C** (Intersection of A and C):
- Find common elements in sets A and C.
- A ∩ C = {2, 4, 6, 8, 10, 12, 14} ∩ {2, 3, 5, 7, 11, 13, 17, 19} = {2}.

(f) **B ∩ C** (Intersection of B and C):
- Find common elements in sets B and C.
- B ∩ C = {3, 6, 9, 12, 15, 18} ∩ {2, 3, 5, 7, 11, 13, 17, 19} = {3}.

### Summary of Results
- (a) A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 18}
- (b) A ∪ C = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 19}
- (c) B ∪ C = {2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}
- (d) A ∩ B = {6, 12}
- (e) A ∩ C = {2}
- (f) B ∩ C = {3}