🧠 Logic and Sets Formulas

🎯 Set Operations

Union

$$A \cup B = {x : x \in A \text{ or } x \in B}$$

Usage: Combine elements from both sets Micro-example: ${1, 2, 3} \cup {3, 4, 5} = {1, 2, 3, 4, 5}$

Intersection

$$A \cap B = {x : x \in A \text{ and } x \in B}$$

Usage: Find elements common to both sets Micro-example: ${1, 2, 3} \cap {3, 4, 5} = {3}$

Complement

$$A’ = {x : x \notin A}$$

Usage: Find elements not in set A (relative to universal set) Micro-example: If universal set is ${1, 2, 3, 4, 5}$ and $A = {1, 3}$, then $A’ = {2, 4, 5}$

Difference

$$A - B = {x : x \in A \text{ and } x \notin B}$$

Usage: Find elements in A but not in B Micro-example: ${1, 2, 3} - {3, 4, 5} = {1, 2}$

🧮 Counting Formulas

Two-Set Inclusion-Exclusion

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Usage: Count elements in union of two sets Micro-example: If $|A| = 10$, $|B| = 15$, and $|A \cap B| = 3$, then $|A \cup B| = 10 + 15 - 3 = 22$

Three-Set Inclusion-Exclusion

$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$

Usage: Count elements in union of three sets Micro-example: If $|A| = 20$, $|B| = 18$, $|C| = 15$, $|A \cap B| = 8$, $|A \cap C| = 6$, $|B \cap C| = 5$, and $|A \cap B \cap C| = 3$, then $|A \cup B \cup C| = 20 + 18 + 15 - 8 - 6 - 5 + 3 = 37$

🔗 Set Properties

Commutative Property

$$A \cup B = B \cup A \text{ and } A \cap B = B \cap A$$

Usage: Order doesn’t matter in union and intersection Micro-example: ${1, 2} \cup {3, 4} = {3, 4} \cup {1, 2}$

Associative Property

$$(A \cup B) \cup C = A \cup (B \cup C) \text{ and } (A \cap B) \cap C = A \cap (B \cap C)$$

Usage: Grouping doesn’t matter in union and intersection Micro-example: $({1, 2} \cup {3, 4}) \cup {5, 6} = {1, 2} \cup ({3, 4} \cup {5, 6})$

Distributive Property

$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \text{ and } A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$

Usage: Distribute operations across parentheses Micro-example: ${1, 2} \cap ({2, 3} \cup {3, 4}) = ({1, 2} \cap {2, 3}) \cup ({1, 2} \cap {3, 4})$

🎲 Logical Operators

Conjunction (And)

$$p \land q$$

Usage: Both p and q must be true Truth table: Only true when both p and q are true

Disjunction (Or)

$$p \lor q$$

Usage: At least one of p or q must be true Truth table: True when at least one is true

Negation (Not)

$$\neg p$$

Usage: Opposite truth value of p Truth table: True when p is false, false when p is true

Implication (If-Then)

$$p \to q$$

Usage: If p is true, then q must be true Truth table: False only when p is true and q is false

💡 Quick Reference

⚠️ Common Pitfalls

• Union vs Intersection: $\cup$ means “or”, $\cap$ means “and”
• Inclusion-Exclusion: Always subtract intersections
• Complement: Relative to universal set, not empty set
• Logical Operators: “Or” is inclusive, not exclusive
• Truth Tables: Check all combinations systematically