📘 set theory

1. **Stating the problem:** We have 96 students in total. The numbers of students participating in different events are: - Field events (F): 15

1. **State the problem:** We are given two finite sets $A$ and $B$ with $n(A) = 40$, $n(B) = 38$, and $n(A \cup B) = 60$. We need to find $n(A \cap B)$, the number of elements in t

1. **Problem Statement:** We have three subsets of whole numbers from 1 to 20:

1. **Problem statement:** Given sets with the following values: $|A|=55$, $|B|=40$, $|C|=80$, $|A \cap B|=20$, $|A \cap B \cap C|=17$, $|B \cap C|=24$, and $|A \cup C|=100$, find:

1. Problem: Find the elements and cardinality of $A \cap B$. Step 1: Identify $A = \{a,b,c,e,f\}$ and $B = \{b,c,d,e,f\}$.

1. **State the problem:** We have a universal set $\xi$ and three sets $C$, $D$, and $E$ with given numbers of elements in each region of their Venn diagram.

1. **State the problem:** We are given the cardinalities of various intersections and unions of three sets $X$, $Y$, and $Z$ inside a universal set $\xi$ with $n(\xi) = 62$. We nee

1. **State the problem:** We have three sets representing students who sat exams in Physics (P), Chemistry (C), and Biology (B). Given are the numbers of students in each subject a

1. The problem asks to find the number of elements in the intersection of two sets A and B given some values. 2. Given: $n(A) = 27$, $n(B) = 25$, $n(A \cap B) = x$, and $n(A \cup B

1. **Problem statement:** We have 50 students taking at least one of Math, Physics, Chemistry.

1. **Problem statement:** Among 100 students, (75 - x) study physics, (50 - x) study accounts, and 25 study neither. No student studies both subjects. Find the value of $x$, the nu

1. **Problem Statement:** Among 100 students, $(75 - x)$ study physics, $(50 - x)$ study accounts, and 25 study neither. No student studies both subjects. Find $x$, the number of s

1. **Problem Statement:** Given the universal set $e = \{1,2,3,4,5,6,7\}$, set $A$ as even numbers, and set $B$ as prime numbers. 2. **Identify sets:**

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. $