📘 set theory

1. Statement of the problem: Given the universal set $\Omega = \{a,b,c,d,e,f,g\}$ and the sets $A=\{a,b,c,d,e\}$, $B=\{a,c,e,g\}$, $C=\{b,e,f,g\}$, find (i) $A \cup B$, (ii) $B \ca

1. **Problem Statement:** List out the members of the sets given. 2. **Set a:** $P = \{1, 2, 3, \ldots, 13\}$ means all integers starting from 1 up to 13.

1. The problem is to create a Venn diagram, which is a visual tool used to show relationships between different sets. 2. Venn diagrams typically consist of overlapping circles, eac

1. **Problem Statement:** We have a survey of 600 investors. - 380 invested in stocks.

1. **Problem Statement:** Find the following from the Venn diagram with sets A, B, C, and universal set 𝓔:

1. **Problem statement:** Given a universal set $E$ and three sets $A$, $B$, and $C$ with elements distributed as follows: - $6$ elements only in $A$

1. The problem is to solve the equation or expression involving the term "set math". Since the user input is ambiguous, we interpret "set math" as a request to explain or solve a p

1. **Problem Statement:** Find the number of relations from a set $A$ to itself if $A$ has $n$ elements. 2. **Formula and Explanation:** A relation from set $A$ to set $A$ is a sub

1. The problem is to simplify the expression $A - (B \cap C)$. 2. This expression represents the set difference between set $A$ and the intersection of sets $B$ and $C$.

1. **Problem statement:** Find the smallest possible number of people who speak both Spanish and French, and the largest possible number of people who speak neither Spanish nor Fre

1. **Problem Statement:** Explain and prove De Morgan's Laws in set theory.

1. The problem states that Preety is a student in Year 12 and that Preety does not study French, i.e., Preety \notin F. 2. Given the sets:

1. The problem states that $\xi$ is the set of students in year 12. 2. $G$ is the set of students who study German.

1. The problem asks to find $n(A)$ where $A = \{1, 2, 3, 4, 5, 6\}$. 2. The notation $n(A)$ represents the number of elements in the set $A$.

1. The problem asks: If $B = \{x: x \text{ is an integer between } 1 \text{ and } 2\}$, list the members of set $B$. 2. To find the members of $B$, we need to identify all integers

1. **State the problem:** Given sets $A = \{1, 2, 3\}$, $B = \{2, 3, 5\}$, and $C = \{2, 5\}$, find the Cartesian product $(A - B) \times (B - C)$. 2. **Recall the definitions:**

1. **Problem statement:** (i) Prove that if $A \subseteq B$ then $P(A) \subseteq P(B)$.

1. **Problem Statement:** Find the intersection $A \cap B$ of the sets $A$ and $B$ given as: $$A = \{a:\{1,2,3,4,5\}, b:\{1,2,3\}, c:\{0,1,2,3\}, d:\{3\}\}$$

1. The problem asks to shade the set corresponding to the formula $ (A \setminus B) \cap \overline{C} $. 2. The formula $ (A \setminus B) \cap \overline{C} $ means elements in $A$

1. **Problem:** Given sets $A = \{a, b, c, d, e\}$ and $B = \{a, b, c, d, e, f, g, h\}$, find: (a) $A \cup B$

1. **Problem:** In a class of 80 students, every student studies Economics and Geography on both. If 65 students study Economics and 50 study Geography, how many study both subject