📚 Key Set Operation Concepts

• Union (E₁ ∪ E₂): All outcomes in E₁ or E₂ or both.
• Intersection (E₁ ∩ E₂): Outcomes in both E₁ and E₂.
• Complement of E₁ (E₁′): Outcomes in S but not in E₁.
• Complement of E₂ (E₂′): Outcomes in S but not in E₂.
• Empty Set (∅): A set with no outcomes. Example: E₁ ∩ E₄ if E₄ = {10, 11}.
• Full Set (S): The entire sample space. S = {1,2,3,4,5,6,7,8,9}.

🧪 Engineering Example – COVID Test Results

Scenario: Biomedical engineers run 2 COVID tests on 9 patients.

• E₁: Positive on test A = {2, 4, 6, 8, 9}
• E₂: Positive on test B = {1, 2, 3, 4, 5, 7}

Sample space (S): All 9 patients = {1,2,3,4,5,6,7,8,9}

These set operations help evaluate test agreement, sensitivity, and missed cases.

📚 Key Set Operation Concepts

• Union (E₁ ∪ E₂): All outcomes in E₁ or E₂ or both.
• Intersection (E₁ ∩ E₂): Outcomes in both E₁ and E₂.
• Complement of E₁ (E₁′): Outcomes in S but not in E₁.
• Complement of E₂ (E₂′): Outcomes in S but not in E₂.
• Empty Set (∅): A set with no outcomes. Example: E₁ ∩ E₄ if E₄ = {10, 11}.
• Full Set (S): The entire sample space. S = {1,2,3,4,5,6,7,8,9}.

🧪 Engineering Example – COVID Test Comparison

Scenario: Engineers are evaluating two different COVID tests used on 9 patients. Each test reports “Positive” or “Negative.”

• Test A (E₁): Set of patients who tested positive with brand A = {2, 4, 6, 8, 9}
• Test B (E₂): Set of patients who tested positive with brand B = {1, 2, 3, 4, 5, 7}

Using set operations:

• E₁ ∪ E₂ (Union): All patients who tested positive on either test → used to measure total positivity risk
• E₁ ∩ E₂ (Intersection): Patients who tested positive on both tests → good for test validation agreement
• E₁′: Patients who tested negative on test A → important for identifying false negatives

This type of analysis helps biomedical engineers evaluate test accuracy and overlap in field trials.

📝 Exercise – Apply Set Logic

Using the following sets:

• S = {1,2,3,4,5,6,7,8,9}
• E₁ = {2,4,6,8,9}
• E₂ = {1,2,3,4,5,7}

Questions:

1. What is E₁ ∪ E₂?
2. What is E₁ ∩ E₂?
3. What is the complement of E₁ (E₁′)?
4. What is the complement of E₂ (E₂′)?
5. What is E₁ ∩ E₂′?
6. What is E₁ ∪ E₂′?