📘 set theory

1. **Stating the problem:** We have three newspapers M (Mirror), C (Citizen), and T (Times). Given:

1. The problem involves three sets of people reading different newspapers: Mirror (M = 65), Citizen (C = 45), and Times (T = 39). Given the numbers who read combinations and none,

1. The problem asks to write the relation $R1 = \{(x,y) : x < y\}$.\n\n2. This relation consists of all pairs $(x,y)$ where the first element $x$ is strictly less than the second e

1. Problem: In a competition, medals were awarded in three categories: dance (36 medals), dramatics (12 medals), and music (18 medals). Total persons awarded = 45. Exactly 4 person

1. **State the problem:** We have a class of 32 students with the following information about subjects offered:

1. Let's state the problem clearly: If a set $S$ has $n$ elements, we want to prove that its power set $\mathcal{P}(S)$ has $2^n$ elements. 2. Recall that the power set of $S$ is t

1. Problem: Show $A \cup B$ by Venn diagram for: (i) Disjoint sets: Two circles representing $A$ and $B$ that do not overlap.

1. **Problem:** Show $A \cup B$ by Venn diagram in different cases. 1.1. When $A$ and $B$ are disjoint sets, $A \cup B$ is the entire area covered by circles $A$ and $B$ without an

1. The problem asks to simplify the expression $(A-B) \cup B$. 2. Recall that $A-B$ means all elements in $A$ that are not in $B$.

1. The problem asks us to construct a bijection (a one-to-one and onto function) from the set of natural numbers $\mathbb{N}$ to the set of odd natural numbers $\mathbb{O}$. This b

1. **Set operations with given sets:** The universal set is $$U = \{a,b,c,d,e,f,g,h,i,j,k,l,m,n\}$$

1. Let's start by understanding what a Venn diagram is. 2. A Venn diagram is a visual tool used to show the relationships between different sets.

1. The problem here is to understand the basics of set theory and its fundamental concepts. 2. A set is a collection of distinct objects, called elements. For example, $A = \{1, 2,

1. **Problem statement:** Given sets $$A = \{ 2, \{3\}, \{2,3\}, 4, \{3,4\} \}$$ and

**Exercise 2: Relation on $\mathbb{R}^3$** Given the binary relation $\mathcal{R}$ on $\mathbb{R}^3$ defined by:

1. The set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ is the set of counting numbers up to 10. 2. The set $C$ is the set of multiples of 3 less than 10, so $C = \{3, 6, 9\}$. The elem

1. **Stating the problem:** We have three sets representing housewives using detergents A, B, and C, with given numbers and intersections. We want to find: a) The number of housewi

1. Statement: We have a total of 140 people. 15 of them like neither TV nor Radio.

1. Let's start with the **definition of a set**: A set is a collection of distinct objects, considered as an object in its own right. 2. Sets are usually denoted by capital letters

1. Олонлогуудын багц нөхцлийг тодорхойлё.\n Нөхцөлүүд: $$\overline{A} \cup \overline{B} = \emptyset,$$ $$X \Delta A = \emptyset,$$ $$B \setminus A \neq \emptyset.$$\n

1. **Problem 1:** In a class of 40 students, 18 are in the Math Club, 15 are in the Science Club, and 7 belong to both clubs. Find how many students are in either club and how many