📘 set theory

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

1. Let's start by stating the problem: We want to understand the difference between a proper set and a subset. 2. A subset is a set where every element of set A is also in set B. W

1. The problem asks to explain why the three brackets indicate nested sets and their importance in set theory. 2. In set theory, a single pair of brackets $\{ \}$ denotes a set con

1. The problem asks: What are the 3 brackets for in the set $B = \{\{\{1, 4, 5, 3, 1\}\}\}$? 2. Each pair of brackets represents a level of set containment.

1. The problem asks: What does set B mean in Problem 4? 2. From Problem 4, set B is defined as:

1. **Problem Statement:** We have 96 students in total. The numbers of students participating in different events are: - Long distance races (L): 50

1. The question asks about the meaning of set B from problem 4. 2. Since the original problem 4 is not provided here, I cannot directly define set B.

1. The problem asks to draw a Venn diagram concerning the answer, but no specific sets or elements are provided. 2. A Venn diagram visually represents the relationships between dif

1. **Stating the problem:** We have 96 students in total. The numbers of students participating in different events are: - Field events (F): 15