📘 set theory

1. **Problem statement:** Given sets with operations and elements:

1. مسئله را بیان می‌کنیم: باید مجموعه $A-B$ را پیدا کنیم که $A=\{x\in \mathbb{R} \mid -1-4 \leq x \leq -1-3\}$ و $B=\{n \in \mathbb{Q} \cap (-,f) \cap \sum n\}$ است. 2. ابتدا باید

1. **Problem Statement:** Identify which set operation corresponds to the shaded region in the given Venn diagram involving sets $A$, $B$, and universal set $U$. 2. **Recall Set Op

1. **Problem:** Find the union of sets $A$ and $B$. **Step 1:** Recall the union formula: $$A \cup B = \{x : x \in A \text{ or } x \in B\}$$

1. **Problem statement:** Given sets and their cardinalities, find possible elements for sets B, C, a value for x in the complement of A, and calculate $n(D \cup C)$. 2. **Recall d

1. **Problem (a):** Find the cardinality of the largest possible set $C$ such that $C \subseteq A$ and $C \subseteq B$ where $A=\{1,6,9,10,11\}$ and $B=\{2,5,7,8,12\}$. 2. **Step 1

1. **State the problem:** We need to shade the complement of the intersection of sets $C$ and $A$, denoted as $(C \cap A)'$. 2. **Recall definitions:**

1. **Problem Statement:** We have a survey of 200 students about reading three newspapers: Imvaho (I), New Times (N), and Daily Times (D). Given data:

1. **Problem statement:** We are given the number of students enrolled in commerce only, mathematics only, both commerce and mathematics, and others. We need to find the total numb

1. **Problem Statement:** We have a survey of 500 students with percentages liking football (F), hockey (H), and basketball (B). Given data:

1. **State the problem:** We need to verify and explain the set identities:

1. **State the problem:** We are given two sets $A = \{1, \{2\}, \{1,2\}\}$ and $B = \{1, \{1,2,3\}\}$. We need to find the symmetric difference $A \Delta B$. 2. **Recall the formu

1. **Problem Statement:** Given three sets:

1. **Stating the problem:** We are given sets $A$, $B$, $S_0$, $S_1$, and $S_r$ with the relations $A \subset S_0$, $S_1 = X \setminus B$, and the chain of inclusions $$\overline{A

1. **Problem:** Given two sets $A$ and $B$ where $A$ is countable and $B$ is uncountable, prove that $(A - B) \times (A \cap B)$ is countable. 2. **Recall definitions and propertie

1. **Problem Statement:** We have three subsets of whole numbers from 1 to 20: - $F = \{2, 4, 6, 8, 10, 12, 14, 18\}$

1. **Problem statement:** We have a multi-level car park with 100 spaces. Let $D$ represent the set of spaces reserved for disabled persons. We want to write the set notation for t

1. **Problem statement:** Given sets A, B, and C with the following cardinalities: $n(A) = 18$, $n(B) = 21$, $n(C) = 22$, $n(A \cap B) = 9$, $n(A \cap C) = 7$, $n(B \cap C) = 11$,

1. **Problem:** Identify which of the given sets is a null set. 2. **Recall:** A null set (empty set) is a set with no elements.

1. **State the problem:** We have 50 children in total. 28 attend piano lessons, 17 attend flute lessons, and 12 attend neither. We need to find how many attend only piano lessons.

1. **State the problem:** We want to find sets $A$, $B$, and $C$ such that: - $A \subseteq B$ (A is a subset of B)