📘 set theory

1. **Problem statement:** Given sets $U=\{a,b,c,d,e\}$, $A=\{a,b,c\}$, and $B=\{b,c,d,e\}$, verify: (i) $ (A \cup B)' = A' \cap B' $

1. **State the problem:** We have three sets defined with conditions on $x$ and three values a=5, b=8, c=4. We want to find which option (A, B, C, or D) correctly matches the value

1. **State the problem:** We have 120 students surveyed about their preferences for three brands of mineral water: Voltaic, Silver, and Deep. Given the numbers who like each brand

1. **State the problem:** We have a survey of 120 college students who like three brands of mineral water: Voltic, Silver, and Deep. Given the numbers of students who like each bra

1. **State the problem:** There are 100 students in a class. 55 study physics, 50 study French, and 30 study geography. 44 study exactly two of the subjects. We need to find how ma

1. **Problem statement:** There are 90 members in a sports club playing at least one of tennis, football, and volleyball. Given: - Tennis and football players: 10

1. **State the problem:** We have 80 senior five students who passed Math (M). Among them, 45 passed Physics (P), 60 passed Chemistry (C), 5 passed Biology (B), and 5 passed only M

1. The problem is to understand the meaning of the symbol \subseteq in mathematics. 2. The symbol \subseteq means "is a subset of or equal to" in set theory.

1. مسئله: بررسی صحت یا نادرستی گزاره‌های داده شده در مبانی نظریه مجموعه‌ها. 2. برای هر گزاره، باید با استفاده از تعاریف مجموعه‌ها، عضویت ($\in$)، زیرمجموعه ($\subseteq$) و عملیات م

1. **Problem:** Find the intersection of sets $A$ and $B$ where $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 3, 5, 7, 9\}$. 2. **Formula:** The intersection of two sets $A$ and $B$, denot

1. The problem is to find the set represented by $A' \cap B$, which means the intersection of the complement of set $A$ and set $B$. 2. Recall that $A'$ (or $A^c$) is the complemen

1. The problem involves finding the symmetric difference $A \Delta B$ for given sets $A$ and $B$. The symmetric difference is defined as the set of elements in either $A$ or $B$ bu

1. The problem asks to find the symmetric difference $A \Delta B$ where $A = \{1, 2, 5\}$ and $B = \{2, 3, 5\}$. 2. The symmetric difference of two sets $A$ and $B$ is defined as t

1. The user requests a Venn diagram with images to understand a problem better. 2. Unfortunately, I cannot create or provide images directly.

1. The problem states that in a survey of 100 people, 60 liked coffee, 40 liked tea, and 20 liked both coffee and tea. 2. We need to find how many people liked only coffee.

1. **Stating the problem:** Given two sets $A$ and $B$ with $A \subseteq B$, determine the sets $A - B$ and $B - A$.

1. مسئله را بیان می‌کنیم: ما باید اشتراک و اجتماع مجموعه‌ها را برای داده‌های زیر پیدا کنیم:

1. The problem is to represent the given information in a Venn diagram. 2. A Venn diagram visually shows the relationships between different sets using overlapping circles.

1. The problem involves three sets: \(U\) is the set of whole numbers from 1 to 10, \(A\) is the set of multiples of 3, \(B\) is the set of factors of 6, and \(C\) is the set of od

1. **Stating the problem:** We have a group of students joining three competitions: Colouring (Mewarna), Singing (Menyanyi), and Cooking (Memasak). The table gives numbers for vari

1. **Problem Statement:** A survey of 120 students found 60 liked cricket (C), 55 liked basketball (B), and 20 liked neither.