📘 set theory

1. **State the problem:** We are given two sets $A = \{1, \{2\}, \{1,2\}\}$ and $B = \{1, \{1,2,3\}\}$. We need to find the symmetric difference $A \Delta B$. 2. **Recall the formu

1. **Problem Statement:** Given three sets:

1. **Stating the problem:** We are given sets $A$, $B$, $S_0$, $S_1$, and $S_r$ with the relations $A \subset S_0$, $S_1 = X \setminus B$, and the chain of inclusions $$\overline{A

1. **Problem:** Given two sets $A$ and $B$ where $A$ is countable and $B$ is uncountable, prove that $(A - B) \times (A \cap B)$ is countable. 2. **Recall definitions and propertie

1. **Problem Statement:** We have three subsets of whole numbers from 1 to 20: - $F = \{2, 4, 6, 8, 10, 12, 14, 18\}$

1. **Problem statement:** We have a multi-level car park with 100 spaces. Let $D$ represent the set of spaces reserved for disabled persons. We want to write the set notation for t

1. **Problem statement:** Given sets A, B, and C with the following cardinalities: $n(A) = 18$, $n(B) = 21$, $n(C) = 22$, $n(A \cap B) = 9$, $n(A \cap C) = 7$, $n(B \cap C) = 11$,

1. **Problem:** Identify which of the given sets is a null set. 2. **Recall:** A null set (empty set) is a set with no elements.

1. **State the problem:** We have 50 children in total. 28 attend piano lessons, 17 attend flute lessons, and 12 attend neither. We need to find how many attend only piano lessons.

1. **State the problem:** We want to find sets $A$, $B$, and $C$ such that: - $A \subseteq B$ (A is a subset of B)

1. The expression $2 |A \cap B|$ involves the cardinality (size) of the intersection of two sets $A$ and $B$. 2. The intersection $A \cap B$ represents all elements that are common

1. **Problem Statement:** In a bootcamp of 60 students:

1. **Problem statement:** Given the set $A = \{\{1, 2, 3\}, \{4, 5\}, \{6, 7, 8\}\}$, we need to: (a) List the elements of $A$.

1. **State the problem:** Determine which of the given sets are null sets (empty sets). 2. **Recall the definition:** A null set (empty set) is a set that contains no elements.

1. **Problem statement:** Prove the Distributive Law: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$. 2. **Formula and rules:** The distributive law in set theory states that i

1. **Problem Statement:** In a city, 20 percent of the population travels by car, 50 percent travels by bus, and 10 percent travels by both car and bus. We need to find the percent

1. The problem asks to illustrate DeMorgan's Law: $$(A \cup B)^C = A^C \cap B^C$$ using Venn diagrams. 2. DeMorgan's Law states that the complement of the union of two sets is equa

1. The problem asks what the notation $\{a, b, c, \ldots, z\}$ represents. 2. This notation is a set notation in mathematics, where elements inside curly braces $\{\}$ represent me

1. **Problem Statement:** We analyze the sets of children who played Fortnite (F), Minecraft (M), and League of Legends (L) based on the given data. 2. **Given Data:**

1. **Problem Statement:** We have 100 members in Majawa village who undertake at least one of three activities: going to school (S), farming (F), and trading (T).

1. **Problem Statement:** Prove the distributive law in sets, which states that for any sets $A$, $B$, and $C$, the following holds: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C