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Questions and Answers
Find the power set of set A = { 1, 2, 3 } and the number of subsets of A.
Find the power set of set A = { 1, 2, 3 } and the number of subsets of A.
The power set of set A, denoted by P(A), is the set of all subsets of A. The power set of A is given by P(A) = { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }. So, the number of subsets of A, denoted by n(A), is 8.
Calculate the symmetric difference between sets P = { 2, 3, 5, 6, 11 } and Q = { 1, 3, 5, 11 }.
Calculate the symmetric difference between sets P = { 2, 3, 5, 6, 11 } and Q = { 1, 3, 5, 11 }.
The symmetric difference between sets P and Q, denoted by P Δ Q, is the set of elements that are in P or Q but not in both. The symmetric difference P Δ Q is given by P Δ Q = {1, 2, 6}.
Verify De Morgan’s laws for set difference using the sets A = { -2, 0, 1, 3, 5 }, B = { -1, 0, 2, 5, 6 }, and C = { -1, 2, 5, 6, 7 }.
Verify De Morgan’s laws for set difference using the sets A = { -2, 0, 1, 3, 5 }, B = { -1, 0, 2, 5, 6 }, and C = { -1, 2, 5, 6, 7 }.
De Morgan’s laws for set difference state that (A - B) U (A - C) = A - (B ∩ C) and (A ∩ B) - C = (A - C) ∪ (B - C). By using the given sets, we can verify these laws to be true.
Verify De Morgan’s laws of set complementation using the sets U = { 4, 7, 8, 10, 11, 12, 15, 16 }, A = { 7, 8, 11, 12 }, and B = { 4, 8, 12, 15 }.
Verify De Morgan’s laws of set complementation using the sets U = { 4, 7, 8, 10, 11, 12, 15, 16 }, A = { 7, 8, 11, 12 }, and B = { 4, 8, 12, 15 }.
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Find the number of students who speak all three languages among 100 students, given the language statistics.
Find the number of students who speak all three languages among 100 students, given the language statistics.
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Study Notes
Power Set and Subsets
- Power set of A = {1, 2, 3} is the set of all subsets of A.
- Power set P(A) includes: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.
- The number of subsets of a set with n elements is calculated as 2^n.
- For set A with 3 elements, the number of subsets is 2^3 = 8.
Symmetric Difference of Sets
- Symmetric difference between sets P = {2, 3, 5, 6, 11} and Q = {1, 3, 5, 11}.
- Symmetric difference includes elements that are in either set but not in both: P Δ Q.
- Result: {1, 2, 6}.
De Morgan's Laws for Set Difference
- Verify De Morgan’s laws using sets A = {-2, 0, 1, 3, 5}, B = {-1, 0, 2, 5, 6}, and C = {-1, 2, 5, 6, 7}.
- The laws state: (A ∩ B') = A - (A ∩ B) and (A ∪ B)' = A' ∩ B'.
- Evaluate both sides to confirm equality for provided sets.
De Morgan's Laws of Set Complementation
- Apply De Morgan’s laws using universal set U = {4, 7, 8, 10, 11, 12, 15, 16}, A = {7, 8, 11, 12}, B = {4, 8, 12, 15}.
- De Morgan’s states: (A ∩ B)' = A' ∪ B' and (A ∪ B)' = A' ∩ B'.
- Check the validity for both sides to ensure correctness using complements.
Language Statistics Among Students
- A scenario involving 100 students speaking three languages.
- Relevant data needed for calculation: number of students speaking each language and overlaps.
- Use provided data to calculate the number of students speaking all three languages.
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Description
Test your knowledge of set theory and combinatorics with this 9th-grade math assignment quiz from Madras Christian College HR. Sec. School. Explore power sets, cardinality, and symmetric differences to enhance your understanding of fundamental mathematical concepts.
