🧠 Logic and Sets Topics#
🎯 Core Subtopics#
Basic Set Operations#
- Union: $A \cup B$ = elements in A or B
- Intersection: $A \cap B$ = elements in both A and B
- Complement: $A’$ = elements not in A
- Micro-example: If $A = {1,2,3}$ and $B = {3,4,5}$, then $A \cup B = {1,2,3,4,5}$ and $A \cap B = {3}$
- Trap: Confusing union and intersection symbols
Venn Diagrams#
- Two-Set Diagrams: Two overlapping circles
- Three-Set Diagrams: Three overlapping circles
- Region Identification: Each region represents a specific combination
- Micro-example: In a two-set diagram, the overlapping region shows $A \cap B$
- Trap: Forgetting to label all regions
Inclusion-Exclusion Principle#
- Two Sets: $|A \cup B| = |A| + |B| - |A \cap B|$
- Three Sets: Add all sets, subtract pairwise intersections, add triple intersection
- Micro-example: If 20 students like math, 15 like science, and 8 like both, then $20 + 15 - 8 = 27$ like at least one
- Trap: Forgetting to subtract the intersection
Set Notation and Language#
- Element of: $x \in A$ means x is in set A
- Subset: $A \subseteq B$ means every element of A is in B
- Proper Subset: $A \subset B$ means A is a subset but not equal to B
- Micro-example: ${1,2} \subseteq {1,2,3}$ and ${1,2} \subset {1,2,3}$
- Trap: Confusing subset and element notation
Logical Statements#
- If-Then: $p \to q$ means “if p then q”
- And: $p \land q$ means “p and q”
- Or: $p \lor q$ means “p or q”
- Not: $\neg p$ means “not p”
- Micro-example: “If it rains, then the ground is wet” is $p \to q$ where p = “it rains” and q = “ground is wet”
- Trap: Confusing “or” (inclusive) with “exclusive or”
Truth Tables#
- Two Variables: 4 rows (TT, TF, FT, FF)
- Three Variables: 8 rows (TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF)
- Micro-example: For $p \land q$, only row TT gives true result
- Trap: Missing rows or wrong truth values
🚨 Common Traps#
- Union vs Intersection: Mixing up $\cup$ and $\cap$ symbols
- Inclusion-Exclusion: Forgetting to subtract intersections
- Venn Diagram Regions: Missing or double-counting regions
- Set Notation: Confusing $\in$ and $\subseteq$
- Logical Operators: Mixing up “and” and “or”
💡 Quick Tips#
- Venn Diagrams: Draw them for any set problem
- Inclusion-Exclusion: Always subtract intersections
- Set Notation: $\in$ for elements, $\subseteq$ for subsets
- Truth Tables: Check all combinations systematically
- Complement: Remember it’s relative to the universal set