📘 set theory

1. **Problem statement:** We have 15 students studying mathematics, 8 studying physics, 6 studying chemistry, and 3 studying all three subjects. We want to prove that 27 or more st

1. **State the problem:** A company needs 30 programmers for system programming and 40 for application programming. They appoint 55 programmers in total. We need to find how many p

1. **Problem statement:** In a class of 52 students, 30 study C++, 28 study Pascal, and 13 study both languages. We need to find how many students study at least one language and h

1. Problem: Find $n[(A \times B) \cup (A \times C)]$ given $n(A) = 2$ and $n(B \cup C) = 3$. 2. Formula: For sets $A$, $B$, and $C$, the Cartesian product $A \times (B \cup C) = (A

1. **Problem:** Prove the set identity $$A \cap (B - C) = (A \cap B) - (A \cap C)$$. 2. **Recall definitions:**

1. Problem: Verify the set identity \((A \cup B) \cap C = (A \cap C) \cup (B \cap C)\). 2. This is a distributive law in set theory. The formula states that the intersection distri

1. مسئله: تعیین صحت عبارات داده شده درباره مجموعه‌ها و شمارش انواع نگاشت‌ها. 2. برای قسمت (الف): اگر $|A| = n + 1$، باید بررسی کنیم که آیا $|B|$ کامل است یا خیر. این جمله نیاز به ت

1. The problem defines two sets: - Set A: $A = \{x \in \mathbb{R} : |x| < 1/2\}$, which means all real numbers $x$ whose absolute value is less than $1/2$.

1. **State the problem:** We want to prove that the mapping $\pi : x \to x/\sim$ is onto. 2. **Understanding the mapping:** The function $\pi$ maps an element $x$ from a set $X$ to

1. **Problem statement:** Given sets $X = \{2, 3\}$, $Y = \{3, 4\}$, and $Z = \{z, y, m\}$ (assuming $Z$ is a set with elements $z, y, m$), find: 1) $Z \times (X \cap Y)$

1. **Problem Statement:** Given the universal set $U = \{a, b, c, d, e, f, g, h\}$ and set $B = \{a, b, c, f\}$, we are asked to:

1. **Stating the problem:** We want to understand when two finite sets $A$ and $B$ are called equivalent or equinumerous (equisovalent). 2. **Definition:** Two finite sets $A$ and

1. **Problem Statement:** Prove that the intersection of sets $A$ and $B$ is commutative, i.e., $A \cap B = B \cap A$. 2. **Definition of Intersection:** The intersection of two se

1. **Problem i:** Given $n(A)=20$, $n(B)=35$, and $n(A \cup B)=45$, find $n(A \cap B)$. 2. **Formula:** For any two sets $A$ and $B$, the cardinality of their union is given by

1. **State the problem:** Given three sets A, B, and C with sizes $n(A)=40$, $n(B)=30$, and $n(C)=35$, and a Venn diagram with intersections labeled as follows: - $12$ in $A$ only

1. مسئله: دو مجموعه داده شده‌اند: $$ B = \{ x \mid x \in \mathbb{Z}, x^{7} - 1 = 0 \} $$

1. مسئله: با توجه به تعریف مجموعه‌ها \( A = \{x \mid x \in W, x - 1 < 7\} \) و \( B = \{x \mid x \in Z, x^2 - 1 = -1\} \)

1. The problem is to understand and represent the set expression $A \cap (B \cap C^\prime)^\prime$ on a Venn diagram. 2. Let's break down the expression step-by-step:

1. সমস্যাটি হলো: ৩০ জন শিক্ষার্থীর মধ্যে ১৮ জন ফুটবল খেলে, ১৪ জন ক্রিকেট খেলে, এবং ৫ জন কিছুই খেলে না। আমাদের জানতে হবে কতজন শিক্ষার্থী উভয় খেলা খেলছে। 2. সূত্র:

1. The problem is to understand what an improper subset means in set theory. 2. A subset $A$ of a set $B$ is a set where every element of $A$ is also an element of $B$.

1. The problem is to understand what an improper subset is. 2. In set theory, a subset $A$ of a set $B$ is called an improper subset if $A$ is exactly equal to $B$.