📐 Sets
Mathematics | FirstInTest
| Concept/Law | Formula/Notation | Notes/Conditions |
|---|---|---|
| Cardinal Number | $n(A)$ | For a finite set A |
| Closed Interval | $[a, b] = \{x: x \in \mathbb{R} \text{ and } a \le x \le b\}$ | For $a, b \in \mathbb{R}$ with $a < b$ |
| Open Interval | $(a, b) = \{x: x \in \mathbb{R} \text{ and } a < x < b\}$ | For $a, b \in \mathbb{R}$ with $a < b$ |
| Semi-Open/Closed Interval | $[a, b) = \{x: x \in \mathbb{R} \text{ and } a \le x < b\}$ | For $a, b \in \mathbb{R}$ with $a < b$ |
| Semi-Open/Closed Interval | $(a, b] = \{x: x \in \mathbb{R} \text{ and } a < x \le b\}$ | For $a, b \in \mathbb{R}$ with $a < b$ |
| Total Number of Subsets | $2^n$ | For a finite set with $n$ elements |
| Power Set of A | $P(A)$ | Collection of all subsets of A |
| Union of two sets A and B | $A \cup B = \{x: x \in A \text{ or } x \in B\}$ | — |
| Intersection of two sets A and B | $A \cap B = \{x: x \in A \text{ and } x \in B\}$ | — |
| Difference of two sets A and B | $A - B = \{x: x \in A \text{ and } x \notin B\}$ | Similarly for $B - A$ |
| Symmetric Difference of two sets A and B | $A \Delta B = (A - B) \cup (B - A)$ | — |
| Complement of set A | $A'$ or $\bar{A}$ | Elements of universal set U not in A |
| Idempotent Laws | $A \cup A = A$ and $A \cap A = A$ | For any set A |
| Identity Laws | $A \cup \emptyset = A$ and $A \cap U = A$ | For any set A, $\emptyset$ is null set, U is universal set |
| Commutative Laws | $A \cup B = B \cup A$ and $A \cap B = B \cap A$ | For any sets A, B |
| Associative Laws | $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$ | For any sets A, B, C |
| Distributive Laws | $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ | For any sets A, B, C |
| De' Morgan's Laws | $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$ | For any sets A, B |
| Cardinality of Union of Two Sets | $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ | For finite sets A, B |
| Cardinality of Union of Two Disjoint Sets | $n(A \cup B) = n(A) + n(B)$ | For disjoint non-void sets A, B |
| Cardinality of Difference of Two Sets | $n(A - B) = n(A) - n(A \cap B)$ | Also $n(A - B) + n(A \cap B) = n(A)$ |
| Cardinality of Symmetric Difference | $n(A \Delta B) = n(A - B) + n(B - A) = n(A) + n(B) - 2n(A \cap B)$ | For finite sets A, B |
| Cardinality of Union of Three Sets | $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$ | For finite sets A, B, C |
| Number of elements in exactly two sets A, B, C | $n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C)$ | For finite sets A, B, C |
| Number of elements in exactly one set A, B, C | $n(A) + n(B) + n(C) - 2n(A \cap B) - 2n(B \cap C) - 2n(C \cap A) + 3n(A \cap B \cap C)$ | For finite sets A, B, C |
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