📘 set theory

1. **Problem Statement:** Given a Venn diagram with sets A and B containing elements: 3, 6, 15, 2, 10, 5, 7, 11, 13, 17, 20, 16, 8, 4, center 12, and 24 (inside overlap circle), an

1. **State the problem:** Find the values of $n(A)$, $n(B)$, $n(A \cap B)$, $n(E)$, $n(A \cup B)$, and $n(B' \cap A)$ from the given Venn diagram. 2. **Given data from the Venn dia

1. **State the problem:** We are given a Venn diagram with sets A and B inside a universal set E, and we need to find the number of elements in various sets. 2. **Recall definition

1. **State the problem:** Write the set $E = \{2, 4, 6, 8, 10\}$ in rule form. 2. **Formula and explanation:** Rule form expresses a set by a property that its members satisfy, usu

1. مسئله: داده شده است \( M = \mathbb{Z} \) (مجموعه اعداد صحیح) و \( A' = \{2, 1, 5, 6\} \) و \( B = \{2, -1, -2\} \). باید مقادیر \( (A \cup B)' \)، \( (A \cap B)' \)، \( (A \cap

1. The problem involves understanding set theory notation and relationships between sets such as $\mathbb{Q}$ (rationals), $\mathbb{Q}^c$ (complement of rationals), $\mathbb{R}$ (r

1. The problem is to understand what builder notation is in mathematics. 2. Builder notation is a way to describe a set by specifying the properties that its members must satisfy.

1. **Problem:** List the elements of set $X$ where $X = \{P : 2 < P \leq 9; P \text{ is a prime number}\}$ and the universal set is $\{1, 2, 3, \ldots, 10\}$. The options are: A. $

1. **State the problem:** Prove that $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$. 2. **Recall the definitions:**

1. **Problem Statement:** Determine if the relation $R_1 = \{(1, -2), (3, 7), (4, -6), (8, 1)\}$ from set $A = \{1,3,4,8\}$ to set $B = \{-2,7,-6,1,2\}$ is a function. 2. **Definit

1. **Problem statement:** We have 100 people surveyed with activities: jogging (J), swimming (S), and cycling (C). Given data: - $|J|=50$, $|S|=30$, $|C|=35$

1. **Problem statement:** We have a survey of 100 people with the following data: - Joggers (J) = 50

1. **Problem statement:** Define the relations $R_1$ and $R_2$ from set $A = \{2,4,6,8\}$ to set $B = \{1,2,3,4\}$.

1. **Problem:** Given multiple questions, we will solve the first one completely as per instructions. **Question 1:** Given sets $U$, $A$, and $B$ where $A$ and $B$ are subsets of

1. **Problem:** Given sets $U$, $A$, and $B$ where $A$ and $B$ are subsets of $U$, determine which of the following statements are true: i. $A \cup B = A \cap B$

1. مسئله: تعیین درستی یا نادرستی عبارات زیر برای مجموعه $A = \{a, \{b\}, \emptyset \}$. 2. فرمول و قواعد: اگر $X \subseteq Y$ باشد، یعنی هر عضو $X$ در $Y$ نیز وجود دارد.

1. مسئله: تعیین درستی یا نادرستی عبارت‌های عضویت در مجموعه $A = \{a, \{b\}, \emptyset\}$. 2. فرمول و قواعد: برای عضویت $x \in A$، $x$ باید یکی از اعضای مجموعه $A$ باشد. توجه کنید ک

1. **Problem Statement:** We have a universal set $U = \{1, 2, 3, \ldots, 29\}$. Set $A$ consists of odd numbers between 10 and 20.

1. **Problem Statement:** We are given three sets:

1. The problem asks to find the intersection of sets A and B, denoted as $A \cap B$. 2. The intersection of two sets includes all elements that are present in both sets.

1. **Problem:** Find the Cartesian product $A \times B$ where $A = \{1,2,3\}$ and $B = \{a,b,c\}$, defined as $A \times B = \{(a,b) : a \in A, b \in B\}$.\n\n2. **Formula and Expla