📘 set theory

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

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

1. **State the problem:** There are 80 men in a club, each playing at least one game: football, hockey, or baseball. Given: 20 play only football, 19 play only hockey, 22 play only

1. **Problem Statement:** In a country club of 144 people, 61 play football, 65 play baseball, and 72 play hockey. 22 play all three games, and 11 play none. We need to find: (i) H

1. **State the problem:** There are 80 men in a club, each playing at least one game: football, hockey, or baseball. Given: 20 play football only, 19 play hockey only, 22 play base

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

1. **Problem Statement:** In a country club of 144 people, 61 play football, 65 play baseball, and 72 play hockey. 22 play all three games, 11 play none, and an equal number play o

1. **Problem Statement:** There are 80 men in a club, each playing at least one game among football, hockey, and baseball. Given:

1. **State the problem:** We are given a set $B = \{1, 2, 3, 4\}$ and need to find how many subsets can be formed from this set. 2. **Formula used:** The number of subsets of a set