Answer :
Let's solve this problem step-by-step.
### Part 1: Defining the Sets and Their Operations
1. Define the Sets for Part (a):
- Let [tex]\( P = \{1, 2, 3, 4\} \)[/tex]
- Let [tex]\( Q = \{3, 4, 5, 6, 7\} \)[/tex]
2. Find the Sizes of Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( n(P) = 4 \)[/tex] (since [tex]\( P \)[/tex] has 4 elements)
- [tex]\( n(Q) = 5 \)[/tex] (since [tex]\( Q \)[/tex] has 5 elements)
3. Find the Intersection of Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( P \cap Q = \{3, 4\} \)[/tex]
- [tex]\( n(P \cap Q) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \)[/tex]:
- Calculate [tex]\( n(P \cup Q) \)[/tex]:
- [tex]\( P \cup Q = \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( n(P \cup Q) = 7 \)[/tex]
- Verify:
- [tex]\( n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \)[/tex]
- [tex]\( 7 = 4 + 5 - 2 \)[/tex]
- The formula holds true.
### Part 2: Verification for Different Sets
1. Define the Sets for Part (b):
- Let [tex]\( A = \{1, 2, 3, 4\} \)[/tex]
- Let [tex]\( B = \{2, 4, 6, 8\} \)[/tex]
2. Find the Sizes of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( n(A) = 4 \)[/tex]
- [tex]\( n(B) = 4 \)[/tex]
3. Find the Intersection of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A \cap B = \{2, 4\} \)[/tex]
- [tex]\( n(A \cap B) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)[/tex]:
- Calculate [tex]\( n(A \cup B) \)[/tex]:
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)[/tex]
- [tex]\( n(A \cup B) = 6 \)[/tex]
- Verify:
- [tex]\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)[/tex]
- [tex]\( 6 = 4 + 4 - 2 \)[/tex]
- The formula holds true.
Similarly,
1. Define the Sets for Part (c):
- Let [tex]\( C = \{a, c, e, m, n\} \)[/tex]
- Let [tex]\( D = \{m, n, p, q, r\} \)[/tex]
2. Find the Sizes of Sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( n(C) = 5 \)[/tex]
- [tex]\( n(D) = 5 \)[/tex]
3. Find the Intersection of Sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( C \cap D = \{m, n\} \)[/tex]
- [tex]\( n(C \cap D) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(C \cup D) = n(C) + n(D) - n(C \cap D) \)[/tex]:
- Calculate [tex]\( n(C \cup D) \)[/tex]:
- [tex]\( C \cup D = \{a, c, e, m, n, p, q, r\} \)[/tex]
- [tex]\( n(C \cup D) = 8 \)[/tex]
- Verify:
- [tex]\( n(C \cup D) = n(C) + n(D) - n(C \cap D) \)[/tex]
- [tex]\( 8 = 5 + 5 - 2 \)[/tex]
- The formula holds true.
### Venn Diagram Representation
(Represent this in Venn Diagram format if needed, showing overlapping sections for intersections.)
### Final Results in Text Format
- For sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( n(P) = 4 \)[/tex]
- [tex]\( n(Q) = 5 \)[/tex]
- [tex]\( P \cap Q = \{3, 4\} \)[/tex]
- [tex]\( n(P \cap Q) = 2 \)[/tex]
- [tex]\( P \cup Q = \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( n(P \cup Q) = 7 \)[/tex]
[tex]\[ n(P \cup Q) = 7 \][/tex]
[tex]\[ n(P) + n(Q) - n(P \cap Q) = 4 + 5 - 2 = 7 \][/tex]
- For sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( n(A) = 4 \)[/tex]
- [tex]\( n(B) = 4 \)[/tex]
- [tex]\( A \cap B = \{2, 4\} \)[/tex]
- [tex]\( n(A \cap B) = 2 \)[/tex]
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)[/tex]
- [tex]\( n(A \cup B) = 6 \)[/tex]
[tex]\[ n(A \cup B) = 6 \][/tex]
[tex]\[ n(A) + n(B) - n(A \cap B) = 4 + 4 - 2 = 6 \][/tex]
- For sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( n(C) = 5 \)[/tex]
- [tex]\( n(D) = 5 \)[/tex]
- [tex]\( C \cap D = \{m, n\} \)[/tex]
- [tex]\( n(C \cap D) = 2 \)[/tex]
- [tex]\( C \cup D = \{a, c, e, m, n, p, q, r\} \)[/tex]
- [tex]\( n(C \cup D) = 8 \)[/tex]
[tex]\[ n(C \cup D) = 8 \][/tex]
[tex]\[ n(C) + n(D) - n(C \cap D) = 5 + 5 - 2 = 8 \][/tex]
These calculations verify the given formulas.
### Part 1: Defining the Sets and Their Operations
1. Define the Sets for Part (a):
- Let [tex]\( P = \{1, 2, 3, 4\} \)[/tex]
- Let [tex]\( Q = \{3, 4, 5, 6, 7\} \)[/tex]
2. Find the Sizes of Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( n(P) = 4 \)[/tex] (since [tex]\( P \)[/tex] has 4 elements)
- [tex]\( n(Q) = 5 \)[/tex] (since [tex]\( Q \)[/tex] has 5 elements)
3. Find the Intersection of Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( P \cap Q = \{3, 4\} \)[/tex]
- [tex]\( n(P \cap Q) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \)[/tex]:
- Calculate [tex]\( n(P \cup Q) \)[/tex]:
- [tex]\( P \cup Q = \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( n(P \cup Q) = 7 \)[/tex]
- Verify:
- [tex]\( n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \)[/tex]
- [tex]\( 7 = 4 + 5 - 2 \)[/tex]
- The formula holds true.
### Part 2: Verification for Different Sets
1. Define the Sets for Part (b):
- Let [tex]\( A = \{1, 2, 3, 4\} \)[/tex]
- Let [tex]\( B = \{2, 4, 6, 8\} \)[/tex]
2. Find the Sizes of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( n(A) = 4 \)[/tex]
- [tex]\( n(B) = 4 \)[/tex]
3. Find the Intersection of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A \cap B = \{2, 4\} \)[/tex]
- [tex]\( n(A \cap B) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)[/tex]:
- Calculate [tex]\( n(A \cup B) \)[/tex]:
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)[/tex]
- [tex]\( n(A \cup B) = 6 \)[/tex]
- Verify:
- [tex]\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)[/tex]
- [tex]\( 6 = 4 + 4 - 2 \)[/tex]
- The formula holds true.
Similarly,
1. Define the Sets for Part (c):
- Let [tex]\( C = \{a, c, e, m, n\} \)[/tex]
- Let [tex]\( D = \{m, n, p, q, r\} \)[/tex]
2. Find the Sizes of Sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( n(C) = 5 \)[/tex]
- [tex]\( n(D) = 5 \)[/tex]
3. Find the Intersection of Sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( C \cap D = \{m, n\} \)[/tex]
- [tex]\( n(C \cap D) = 2 \)[/tex]
4. Verify the Formula [tex]\( n(C \cup D) = n(C) + n(D) - n(C \cap D) \)[/tex]:
- Calculate [tex]\( n(C \cup D) \)[/tex]:
- [tex]\( C \cup D = \{a, c, e, m, n, p, q, r\} \)[/tex]
- [tex]\( n(C \cup D) = 8 \)[/tex]
- Verify:
- [tex]\( n(C \cup D) = n(C) + n(D) - n(C \cap D) \)[/tex]
- [tex]\( 8 = 5 + 5 - 2 \)[/tex]
- The formula holds true.
### Venn Diagram Representation
(Represent this in Venn Diagram format if needed, showing overlapping sections for intersections.)
### Final Results in Text Format
- For sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- [tex]\( n(P) = 4 \)[/tex]
- [tex]\( n(Q) = 5 \)[/tex]
- [tex]\( P \cap Q = \{3, 4\} \)[/tex]
- [tex]\( n(P \cap Q) = 2 \)[/tex]
- [tex]\( P \cup Q = \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]
- [tex]\( n(P \cup Q) = 7 \)[/tex]
[tex]\[ n(P \cup Q) = 7 \][/tex]
[tex]\[ n(P) + n(Q) - n(P \cap Q) = 4 + 5 - 2 = 7 \][/tex]
- For sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( n(A) = 4 \)[/tex]
- [tex]\( n(B) = 4 \)[/tex]
- [tex]\( A \cap B = \{2, 4\} \)[/tex]
- [tex]\( n(A \cap B) = 2 \)[/tex]
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 8\} \)[/tex]
- [tex]\( n(A \cup B) = 6 \)[/tex]
[tex]\[ n(A \cup B) = 6 \][/tex]
[tex]\[ n(A) + n(B) - n(A \cap B) = 4 + 4 - 2 = 6 \][/tex]
- For sets [tex]\( C \)[/tex] and [tex]\( D \)[/tex]:
- [tex]\( n(C) = 5 \)[/tex]
- [tex]\( n(D) = 5 \)[/tex]
- [tex]\( C \cap D = \{m, n\} \)[/tex]
- [tex]\( n(C \cap D) = 2 \)[/tex]
- [tex]\( C \cup D = \{a, c, e, m, n, p, q, r\} \)[/tex]
- [tex]\( n(C \cup D) = 8 \)[/tex]
[tex]\[ n(C \cup D) = 8 \][/tex]
[tex]\[ n(C) + n(D) - n(C \cap D) = 5 + 5 - 2 = 8 \][/tex]
These calculations verify the given formulas.
