Practice Questions

List all the subsets of the set $C = \{x, y\}$.

Recall the name of the law represented by the property: $A \cup (B \cup C) = (A \cup B) \cup C$.

Write the set $A = \{x : x \text{ is a prime factor of } 154\}$ in roster form.

Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and a set $A = \{x : x \text{ is a multiple of 3}\}$. Calculate $A'$.

If $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$, and $C = \{5, 6, 7, 8\}$, calculate the set $(A \cup B) \cap (B \cup C)$.

Formulate the set $A = \{1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \dots\}$ in set-builder form.

Identify if the following collection is a set or not. Justify your answer: 'The collection of the four most beautiful cities in the world.'

Define a set.

If $P = \{a, e, i\}$ and $Q = \{i, o, u\}$, calculate $P \cup Q$.

Evaluate the statement: 'If $A \subset C$ and $B \subset C$, then $A \cup B \subset C$'. If it is true, provide a proof. If it is false, provide a counterexample.

State the definition of a finite set.

For any two sets A and B, formulate a proof to show that A - B = A \cap B'\.

Let set $A$ be the set of letters in the word 'LISTEN' and set $B$ be the set of letters in the word 'SILENT'. Analyze if sets $A$ and $B$ are equal. Justify your answer.

Justify why the collection of 'all difficult problems in this chapter' is not a set, while the collection of 'all questions in Exercise 1.1' is a set.

A student claims that for any non-empty set A with $n$ elements, the number of its proper subsets is $2^n - 2$, arguing that both the empty set $\phi$ and the set A itself must be excluded. Critique this reasoning and provide the correct formula.

Explain the difference between the symbols $\in$ and $\subset$.

Let $X = \{x : x \in \mathbf{N} \text{ and } x \le 5\}$ and $Y = \{x : x \text{ is an even natural number less than } 8\}$. Calculate $X \cap Y$.

Represent the set $B = \{1, 8, 27, 64, 125\}$ in set-builder form.

Let $A = [-3, 5)$ and $B = (1, 7]$ be two intervals on the real number line. Calculate (i) $A \cap B$ and (ii) $A - B$.

Given the sets $A = \{x : x \text{ is an integer, } -2 \le x < 3\}$ and $B = \{y : y \text{ is a whole number, } y < 4\}$. Calculate $(A - B) \cup (B - A)$.

Analyze which of the following pairs of sets are disjoint. (i) $A = \{x : x \text{ is a prime number less than } 10\}$ and $B = \{x : x \text{ is an even integer and } 0 < x < 10\}$. (ii) $C = \{1, 3, 5, 7\}$ and $D = \{2, 4, 6, 8\}$.

List the elements of the set $A = \{x : x \text{ is a letter in the word 'MATHEMATICS'}\}$ in roster form.

Let $U = \{1, 2, 3, 4, 5, 6, 7\}$, $A = \{2, 3, 5\}$, and $B = \{3, 5, 6\}$. Demonstrate that $(A \cup B)' = A' \cap B'$.

Let the universal set be $U=\{x: x \in \mathbf{Z} \text{ and } 0 \leq x \leq 10\}$. Let $A=\{x: x \in U \text{ and } x \text{ is a prime number}\}$, $B=\{x: x \in U \text{ and } x \text{ is an even number}\}$, and $C=\{1, 2, 3, 4\}$. Calculate (i) $A' \cap B$ and (ii) $(A \cup C) - B$.

Let set $A = \{x \in \mathbf{Z} : x^2 \le 9\}$ and set $B = \{x \in \mathbf{R} : x^2 - 4x + 3 = 0\}$. Compare the two sets. Determine if $B \subset A$. Calculate $A - B$ and $A \cap B$.

Justify, using properties of sets, that the symmetric difference of two sets A and B, defined as $(A - B) \cup (B - A)$, is equal to $(A \cup B) - (A \cap B)$.

Using the definitions of set operations and elements (i.e., not Venn diagrams), formulate a rigorous proof for De Morgan's Law: (A \cup B)' = A' \cap B'\.

Let A, B, and C be sets. Evaluate the combined conditions $A \cup B = A \cup C$ and $A \cap B = A \cap C$. Formulate a proof to show that these two conditions together imply that $B = C$.

Propose a suitable universal set U for the sets $A = \{x : x \text{ is a prime factor of } 210\}$ and $B = \{x : x \text{ is a solution of } x^2 - 9x + 14 = 0\}$. Justify your choice.

Create a specific example of three non-empty sets A, B, and C such that their pairwise intersections are non-empty, but the intersection of all three is empty.

Let $A = \{x, y, z\}$. The power set of A, denoted $P(A)$, is the set of all subsets of A. Analyze the following statements and determine if they are correct: (i) $\{x, y\} \in P(A)$ (ii) $\{x\} \subset P(A)$.

For any two sets A and B, formulate a proof to show that if the power set of A is a subset of the power set of B, then A is a subset of B. (i.e., $P(A) \subset P(B) \Rightarrow A \subset B$)

Design a proof to show that for any two sets A and B, $A \cup B = (A - B) \cup (B - A) \cup (A \cap B)$ and demonstrate that the three sets on the right-hand side are mutually disjoint.

Justify that the following three statements are equivalent for any sets A and B within a universal set U: (i) $A \subset B$ (ii) $A \cap B' = \phi$ (iii) $A' \cup B = U$

For any three sets A, B, and C, create a proof to show that $A - (B \cup C) = (A - B) \cap (A - C)$.

Let the universal set be $U = \{ \text{all single digit natural numbers} \}$. Let $A = \{ x \in U : x \text{ is a perfect square} \}$ and $B = \{ x \in U : x \text{ is a factor of 12} \}$. First, calculate $A'$ and $B'$. Then, analyze the set $(A \cup B)'$ by applying De Morgan's law and verifying it through direct calculation.

Evaluate whether the statement $(A \cup B) \cap A' = B - A$ is always true for any sets A and B. Justify your answer using set properties.