📘 set theory

1. **State the problem:** We have 136 boys who play rugby, soccer, or hockey. Given the counts for each sport and their pairwise overlaps, we want to find how many boys played all

1. **Stating the problem:** Given the universal set $$U = \{1, 2, 3, ..., 11\}$$, and sets $$A = \{1, 3, 5, 7, 11\}$$ and $$B = \{3, 5, 7, 9\}$$, we want to find:

1. مسئله را بیان می‌کنیم: نوشتن صورت ریاضی و اعضای مجموعه‌های ارائه شده و نمایش اجتماع دو مجموعه A و B. 2. مجموعه اعداد حسابی (W) تعریف شده as

1. The problem asks us to illustrate given information on a Venn diagram. 2. Since no specific sets or elements are provided, we cannot draw a specific Venn diagram.

1. **State the problem:** In a class of 36 students, 19 read biology, 16 read chemistry, and 5 students do not read either subject. Find how many students read both biology and che

1. Stating the problem: Identify which sets among A, B, C, D, E, F, G, H are equal. Step 1: Solve equations for A and B.

1. You requested Venn diagrams for each question. 2. Since no specific mathematical problem involving sets or Venn diagrams was provided, I cannot generate corresponding diagrams o

1. Problem 26: Draw Venn diagrams for the sets A, B, and C with following expressions: a) $A \cap (B \cup \overline{C})$

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,