📘 set theory

1. **Problem statement:** We have a class of 70 students playing football, volleyball, and basketball with given intersection counts. We need to answer several questions about the

1. **State the problem:** Prove that $$A \cap (B - C) = (A \cap B) - (A \cap C)$$ where $$B - C = \{x \mid x \in B \text{ and } x \notin C\}$$. 2. **Rewrite the left-hand side (LHS

1. The problem is to prove the set identity: $$A \cap (B - C) = (A \cap B) - (A \cap C)$$. 2. Recall that the set difference $B - C$ is defined as $\{x \mid x \in B \text{ and } x

1. The problem is to simplify the expression \( \left\{\;\right\} \), which represents an empty set or no elements inside the braces. 2. Since there is nothing inside the braces, t

1. The symbol \cup represents the union operation in set theory. 2. The union of two sets A and B, denoted by $A \cup B$, is the set containing all elements that are in A, or in B,

1. **State the problem:** We have 100 apples. Among them, 20 have worms, 15 have bruises, and 10 have both worms and bruises. We want to find how many apples have neither worms nor

1. **State the problem:** Prove that $ (A \cap C) \setminus ((A \setminus B) \setminus (B \setminus C)) = A \setminus B $ for all sets $A, B, C$. 2. **Recall set theory laws:**

**Problem:** Prove that for all sets $A, B, C$, the expression $B \cap C, (A - B) - (B - C) = A - B$ holds. 1. **Understand the expression:** The expression combines set intersecti

1. **State the problem:** Prove the set equality $ (A - B) - (B - C) = A - B $ for all sets $ A $, and given sets $ B $ and $ C $. 2. **Recall definitions and laws:**

1. **State the problem:** In a class of 20 boys, 16 play soccer, 12 play hockey, and 2 are not allowed to play games. We need to find the number of students who play both soccer an

1. **State the problem:** We have sets \(\xi = \{23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34\}\), \(A = \{\text{even numbers}\}\), \(B = \{23, 29, 31\}\), \(C = \{\text{multiple

1. **State the problem:** We need to determine if the intersection of sets $B$ and $C$ is empty, i.e., if $B \cap C = \emptyset$. 2. **Recall the sets:**

1. **State the problem:** We have three sets:

1. **Stating the Problem**: We are given three sets: - $\xi = \{23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34\}$

1. The user requested a Venn diagram which is a graphical representation and not a math problem with calculations or formulas. 2. As a math tutor, I can provide guidance on set the

1. The problem is to explain and visualize a Venn diagram. 2. A Venn diagram shows relationships between different sets using overlapping circles.

1. Problem: At a luncheon party for 37 people, given various counts of requests for fried rice (F), salad (S), and shredded beef (B), find: (i) the number of people requesting all

1. **State the problem:** We have 120 people visiting a shop. We know: - $\frac{3}{4}$ of them buy neither a coat nor a dress.

1. **Stating the problem:** We have three sets representing professionals working in the morning (M), afternoon (A), and night (N). Given the total counts for each set and their in

1. The problem asks to identify the Venn diagram that correctly represents the set expression $$ (A - B) \cup (B - A) $$. 2. The set $$ (A - B) $$ represents elements that are in s

1. Problem: Determine which sets among A, B, C, D, E, F, G, H are equal given their definitions. Step 1: Find elements of each set.