📘 set theory

1. The problem states that Preety is a student in Year 12 and that Preety does not study French, i.e., Preety \notin F. 2. Given the sets:

1. The problem states that $\xi$ is the set of students in year 12. 2. $G$ is the set of students who study German.

1. The problem asks to find $n(A)$ where $A = \{1, 2, 3, 4, 5, 6\}$. 2. The notation $n(A)$ represents the number of elements in the set $A$.

1. The problem asks: If $B = \{x: x \text{ is an integer between } 1 \text{ and } 2\}$, list the members of set $B$. 2. To find the members of $B$, we need to identify all integers

1. **State the problem:** Given sets $A = \{1, 2, 3\}$, $B = \{2, 3, 5\}$, and $C = \{2, 5\}$, find the Cartesian product $(A - B) \times (B - C)$. 2. **Recall the definitions:**

1. **Problem statement:** (i) Prove that if $A \subseteq B$ then $P(A) \subseteq P(B)$.

1. **Problem Statement:** Find the intersection $A \cap B$ of the sets $A$ and $B$ given as: $$A = \{a:\{1,2,3,4,5\}, b:\{1,2,3\}, c:\{0,1,2,3\}, d:\{3\}\}$$

1. The problem asks to shade the set corresponding to the formula $ (A \setminus B) \cap \overline{C} $. 2. The formula $ (A \setminus B) \cap \overline{C} $ means elements in $A$

1. **Problem:** Given sets $A = \{a, b, c, d, e\}$ and $B = \{a, b, c, d, e, f, g, h\}$, find: (a) $A \cup B$

1. **Problem:** In a class of 80 students, every student studies Economics and Geography on both. If 65 students study Economics and 50 study Geography, how many study both subject

1. Problem: Determine the nature of the set $A = (2,4,6,8,10, \ldots)$. - This set contains even numbers starting from 2 and continues indefinitely.

1. The problem asks to identify the type of set A = {2,4,6,8,10, ...}. 2. This set contains even numbers starting from 2 and continues indefinitely.

1. The problem is unclear as the input is just the symbol \in, which is used in mathematics to denote "element of" in set theory. 2. Typically, \in is used to express that an eleme

1. **Stating the problem:** We have 100 students in total.

1. The problem involves identifying the shaded region in a Venn diagram with three sets $P$, $Q$, and $R$ within the universal set $\xi = P \cup Q \cup R$. 2. The shaded region is

1. مسئله: یافتن اشتراک دو مجموعه $A = (-\infty, 2]$ و $B = (-1, 5]$ است. 2. تعریف اشتراک: اشتراک دو مجموعه $A \cap B$ شامل تمام عناصری است که در هر دو مجموعه وجود دارند.

1. مسئله: مجموعه $A = \{\emptyset, \{1\}, \{2\}, \{1, \{2\}\}, \{\emptyset\}\}$ را در نظر بگیرید. الف) تعداد اعضای مجموعه $A$ را بیابید. 2. برای شمارش اعضای مجموعه، هر عضو را جداگا

1. **State the problem:** We are given two sets $P = \{a, b, c, d, e\}$ and $R = \{t, d, c, b, e\}$. We need to find the set difference $P - R$. 2. **Formula and explanation:** The

1. **Problem:** Compute the union and intersection of given sets and other set operations. Given sets:

1. **Problem:** Compute $A \cup B$ where $A = \{a, b, c, g\}$ and $B = \{d, e, f, g\}$. 2. **Formula:** The union of two sets $A$ and $B$ is defined as $A \cup B = \{x : x \in A \t

1. The problem states: "B is a set of days of the week that starts with A." We need to understand what this means. 2. The days of the week are: Monday, Tuesday, Wednesday, Thursday