📘 set theory

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:

1. **Problem Statement:** We have 100 students selling fruits: 40 sell apples (A), 46 sell oranges (O), 50 sell mangoes (M). 14 sell both apples and oranges, 15 sell both apples an

1. **Problem Statement:** We have 100 students selling fruits: 40 sell apples, 46 sell oranges, 50 sell mangoes. 14 sell both apples and oranges, 15 sell both apples and mangoes, a

1. The problem asks us to identify which of the given sets is finite. 2. A finite set is a set with a limited number of elements. An infinite set has unlimited elements.

1. **Stating the problem:** We need to prove or disprove the set theory statement $A \cap B = A \cup B$.

1. **Problem statement:** Prove using set identities that $$\overline{A} \cup \overline{B} \cup (A \cap B \cap \overline{C}) = \overline{A} \cup \overline{B} \cup \overline{C}$$ 2.

1. The problem is to understand and perform operations on sets. 2. Common set operations include union ($A \cup B$), intersection ($A \cap B$), difference ($A - B$), and complement

1. Let's start by understanding what set theory is. Set theory is a branch of mathematical logic that studies sets, which are collections of objects. 2. A set is usually denoted by

1. The problem is to practice basic set theory operations such as union, intersection, difference, and complement. 2. Important formulas and rules: