Introduction to Elementary Set Theory

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Questions and Answers

Given a universal set $U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$, set $A = {2, 4, 6, 8, 10}$, and set $B = {1, 3, 5, 7, 9}$, determine the complement of $(A \cup B)$ and explain your reasoning.

$(A \cup B) = U$, so $(A \cup B)' = \emptyset$. The union of A and B is the universal set itself because A contains all even numbers from U and B contains all odd numbers from U. Therefore, the complement of their union is the empty set.

Let $A$, $B$, and $C$ be sets. Prove or disprove: If $A \cap B = A \cap C$, then $B = C$. Provide a counterexample if the statement is false.

The statement is false. Counterexample: Let $A = {1, 2}$, $B = {1, 3}$, and $C = {1, 4}$. Then $A \cap B = {1}$ and $A \cap C = {1}$, so $A \cap B = A \cap C$, but $B \neq C$.

Suppose $A$ and $B$ are sets such that $P(A) = P(B)$. What can you conclude about the relationship between $A$ and $B$, and why?

$A = B$. The power set $P(A)$ uniquely determines $A$, as $A$ is the union of all singleton sets in $P(A)$. If $P(A) = P(B)$, then both power sets contain the same singleton sets, implying $A$ and $B$ must be composed of the same elements and are therefore equal.

Let $A$ and $B$ be two non-empty sets. If $A \subseteq B$, what can you say about the relationship between $A \cup B$ and $A \cap B$?

If $A \subseteq B$, then $A \cup B = B$ and $A \cap B = A$. This is because if all elements of A are also in B, then the union of A and B is simply B, and the intersection of A and B is A. Signup and view all the answers

Given three sets $A$, $B$, and $C$, prove or disprove: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

The statement is true. Proof: $x \in A \cup (B \cap C)$ iff $x \in A$ or $x \in (B \cap C)$. This is equivalent to $x \in A$ or ($x \in B$ and $x \in C$), which is the same as ($x \in A$ or $x \in B$) and ($x \in A$ or $x \in C$), which is equivalent to $x \in (A \cup B) \cap (A \cup C)$. Signup and view all the answers

Suppose a survey of 100 people reveals that 60 like apples, 50 like bananas, and 20 like both. How many people like neither apples nor bananas?

30 people like neither apples nor bananas. Using the principle of inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B| = 60 + 50 - 20 = 90$. Therefore, $100 - 90 = 10$ people like at least one of the fruits, and $100-90 = 10$ like neither. Signup and view all the answers

Let $A$ and $B$ be subsets of a universal set $U$. Simplify the expression $(A \cap B) \cup (A \cap B')$ and explain your reasoning.

$(A \cap B) \cup (A \cap B') = A \cap (B \cup B') = A \cap U = A$. The union of a set and its complement ($B \cup B'$) is the universal set $U$, and the intersection of any set with the universal set is the set itself. Signup and view all the answers

If $|A| = m$ and $|B| = n$, where $A$ and $B$ are disjoint sets, what is the cardinality of the power set of $A \cup B$?

$|P(A \cup B)| = 2^{m+n}$. Since $A$ and $B$ are disjoint, $|A \cup B| = |A| + |B| = m + n$. The cardinality of the power set of a set with $k$ elements is $2^k$. Therefore, the cardinality of the power set of $A \cup B$ is $2^{m+n}$. Signup and view all the answers

Consider a universal set $U$ and its subsets $A$ and $B$. If $A \subseteq B$, what is $A \setminus (A \cap B)$?

$A \setminus (A \cap B) = \emptyset$. Since $A \subseteq B$, $A \cap B = A$. Therefore, $A \setminus (A \cap B) = A \setminus A = \emptyset$. Signup and view all the answers

Describe DeMorgan's Laws in the context of set theory. Provide examples for both laws using specific sets.

DeMorgan's Laws state: 1) $(A \cup B)' = A' \cap B'$ (The complement of the union is the intersection of the complements). 2) $(A \cap B)' = A' \cup B'$ (The complement of the intersection is the union of the complements). Example: Let $U = {1, 2, 3, 4, 5}$, $A = {1, 2}$, $B = {2, 3}$. Then $(A \cup B)' = {1, 2, 3}' = {4, 5}$ and $A' \cap B' = {3, 4, 5} \cap {1, 4, 5} = {4, 5}$. Also, $(A \cap B)' = {2}' = {1, 3, 4, 5}$ and $A' \cup B' = {3, 4, 5} \cup {1, 4, 5} = {1, 3, 4, 5}$. Signup and view all the answers

Flashcards

What is a Set?

A well-defined collection of distinct objects, considered as an object in its own right.

What is the Roster Method?

A method to define a set by listing all its elements within curly braces.