📘 set theory

1. **State the problem:** We have 120 students studying Mathematics (M), Economics (E), and Computer Studies (C) with given overlaps. We want to represent this information using a

1. **State the problem:** We have 120 students surveyed about their study of Mathematics (M), Economics (E), and Computer Studies (C). Given the numbers studying each subject and t

1. The problem asks for two difficult problems involving sets. 2. Let's consider the first problem: Given two sets $A$ and $B$, find the set expression for elements that are in $A$

1. **Problem statement:** In a class of 50 students, 30 like math, 25 like science, and 10 like neither. We need to find:

1. **Problem statement:** In a class of 30 students, 30 like Math, 25 like Science, and 10 like neither subject. We need to find: - Number of students who like both subjects.

1. **State the problem:** Determine whether each set C is a subset of set D for the given pairs. 2. **Recall the definition of subset:** A set $C$ is a subset of $D$, written $C \s

1. **Stating the problem:** We have data about 120 students studying three languages: French (F), German (G), and Russian (R). We want to organize this data and understand the numb

1. **Problem statement:** Prove that the complement of the union of sets $A$ and $B$ equals the intersection of their complements, i.e., $$\overline{A \cup B} = \overline{A} \cap \

1. The problem asks for the intersection of two sets $A$ and $B$, where $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$. The intersection of two sets, denoted $A \cap B$, is the set of ele

1. **Problem:** Prove that $A \cap B$ is the empty set given $A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$. 2. **Formula and Rules:** The intersection of two sets $A$ and $B$, denoted $A

1. **State the problem:** There are 150 Grade 7 students. 64 registered in Math Trail, 78 in Amazing Race, and 28 in both events. We need to find how many registered only in Math T

1. The problem asks to draw a Venn diagram for three sets $A$, $B$, and $C$ with the following conditions: - $A \subseteq B$ (set $A$ is a subset of set $B$)

1. The problem is to understand what $n(A)$ means in set theory or probability. 2. $n(A)$ represents the number of elements in the set $A$. It is called the cardinality of the set

1. **State the problem:** We have two sets: \(A\) = multiples of 3, \(B\) = even numbers, and the universal set is \(\{2,3,4,6,8,9,10,12,14,15\}\). 2. **Find \(n(A)\):** Count elem

1. **Problem:** If $C = \{1,3,9\}$, $D = \{3,5,7\}$, and $E = \{3,5,7,9,11\}$, prove using a Venn diagram that $$C \cup (D \cap E) = (C \cup D) \cap (C \cup E)$$

1. נתחיל בפתרון סעיף (א): הבעיה: נתונה קבוצת $A=\mathbb{N}$ ויחס שקילות $R = \{(x,y) \in A^2 : x = y \text{ או } x \cdot y \text{ אי-זוגי}\}$. יש למצוא את קבוצת המנה $A/R$.

1. **Problem Statement:** An agent sells three magazines: Mwananchi, Nipashe, and Daily News. Given the number of customers buying each and the overlaps, we want to represent this

1. The problem is to understand and solve a question involving a Venn diagram, which typically represents sets and their relationships. 2. Venn diagrams use circles to show how dif

1. **Problem statement:** We have three magazines: Mwananchi (M), Nipashe (N), and Daily News (D). Given: - $|M|=70$, $|N|=60$, $|D|=50$

1. **State the problem:** We have 68 students in total participating in sports events: field, track, and swimming. Given counts for various intersections and exclusive groups, we n

1. **Problem statement:** We have 110 members playing at least one of football, basketball, volleyball. Given: