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