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PPSC Lecturer Math Metric Spaces Solved MCQs & Quiz
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PPSC Lecturer Math Metric Spaces Solved MCQs & Quiz
In advanced competitive examinations like the Punjab Public Service Commission (PPSC) and Federal Public Service Commission (FPSC) Lecturer in Mathematics tests, Metric Spaces forms a foundational pillar of Real Analysis and Topology. Concepts covered in standard university reference texts (such as Walter Rudin's Principles of Mathematical Analysis and S.M. Yusuf's Mathematical Methods) are regularly tested. Preparing at a dedicated math learning center requires mastering these rigorous formal definitions and fundamental theorems.
Let $X$ be a non-empty set. A function $d: X \times X \to \mathbb{R}$ is called a metric on $X$ if it satisfies the following four fundamental axioms for all $x, y, z \in X$:
The pair $(X, d)$ is called a metric space. Below is a quick summary table of key metrics, topological ball definitions, and sequence properties frequently asked in PPSC/FPSC exams:
| Metric Concept | Mathematical Formula / Definition | Key Exam Property |
|---|---|---|
| Euclidean Metric ($d_2$) | $$d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}$$ | Standard metric on $\mathbb{R}^n$; complete metric space. |
| Discrete Metric ($d_D$) | $$d_D(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases}$$ | Induces the discrete topology; every set is both open and closed (clopen). |
| Maximum / Chebyshev Metric ($d_\infty$) | $$d_\infty(x, y) = \max_{1 \le i \le n} |x_i - y_i|$$ | Open balls are $n$-dimensional hypercubes. |
| Open Ball $B(x_0, r)$ | $$B(x_0, r) = \{ x \in X : d(x, x_0) < r \}$$ | Forms a base for the topology induced by metric $d$. |
| Diameter of Set $A \subseteq X$ | $$\text{diam}(A) = \sup \{ d(x, y) : x, y \in A \}$$ | $A$ is bounded if and only if $\text{diam}(A) < \infty$. |
Question 1 (Part i): Let $(X, d)$ be a discrete metric space, where $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$. Find the open ball $B(x_0, r)$ centered at $x_0 \in X$ with radius $r = \frac{1}{2}$.
Solution:
By definition, the open ball $B(x_0, r)$ in a metric space $(X, d)$ is defined as:
$$B(x_0, r) = \{ x \in X : d(x_0, x) < r \}$$
Substitute $r = \frac{1}{2}$:
$$B\left(x_0, \frac{1}{2}\right) = \left\{ x \in X : d(x_0, x) < \frac{1}{2} \right\}$$
In a discrete metric space, the metric $d(x_0, x)$ can only take two values: $0$ (when $x = x_0$) or $1$ (when $x \neq x_0$).
Since $0 < \frac{1}{2}$ and $1 \nless \frac{1}{2}$, the condition $d(x_0, x) < \frac{1}{2}$ is satisfied solely when $d(x_0, x) = 0$, which forces $x = x_0$.
Therefore, the open ball contains only the center point $x_0$.
Answer: $$B\left(x_0, \frac{1}{2}\right) = \{x_0\}$$
Question 1 (Part ii): For the same discrete metric space $(X, d)$, find the open ball $B(x_0, r)$ with radius $r = 2$.
Solution:
Using the open ball definition for radius $r = 2$:
$$B(x_0, 2) = \{ x \in X : d(x_0, x) < 2 \}$$
For any point $x \in X$, $d(x_0, x)$ is either $0$ or $1$.
Since both $0 < 2$ and $1 < 2$, every single element $x \in X$ satisfies $d(x_0, x) < 2$.
Thus, every element of $X$ lies inside $B(x_0, 2)$.
Answer: $$B(x_0, 2) = X$$
Question 2: Prove that the function $d^*(x, y) = \frac{|x - y|}{1 + |x - y|}$ defines a valid metric on $\mathbb{R}$.
Solution:
We must verify all four metric axioms for $d^*(x, y)$ on $\mathbb{R}$:
1. Non-negativity: Since $|x - y| \ge 0$ and $1 + |x - y| > 0$, we have $d^*(x, y) \ge 0$.
2. Identity of Indiscernibles:
$$d^*(x, y) = 0 \iff \frac{|x - y|}{1 + |x - y|} = 0 \iff |x - y| = 0 \iff x = y$$
3. Symmetry:
$$d^*(x, y) = \frac{|x - y|}{1 + |x - y|} = \frac{|y - x|}{1 + |y - x|} = d^*(y, x)$$
4. Triangle Inequality: Consider the auxiliary function $f(t) = \frac{t}{1 + t}$ for $t \ge 0$.
Taking the derivative: $f'(t) = \frac{(1+t) - t(1)}{(1+t)^2} = \frac{1}{(1+t)^2} > 0$.
Hence, $f(t)$ is strictly increasing on $[0, \infty)$.
By standard triangle inequality for real numbers, $|x - z| \le |x - y| + |y - z|$. Let $a = |x - y|$ and $b = |y - z|$. Since $f$ is increasing:
$$f(|x - z|) \le f(a + b) = \frac{a + b}{1 + a + b} = \frac{a}{1 + a + b} + \frac{b}{1 + a + b} \le \frac{a}{1 + a} + \frac{b}{1 + b}$$
Substituting back $a$ and $b$:
$$d^*(x, z) \le d^*(x, y) + d^*(y, z)$$
All four axioms hold true.
Answer: $$d^*(x, y) \text{ is a valid metric on } \mathbb{R}$$
Question 3: Let $(X, d)$ be a metric space. Prove the Reverse Triangle Inequality: $|d(x, z) - d(y, z)| \le d(x, y)$ for all $x, y, z \in X$.
Solution:
By the standard triangle inequality for the metric $d$:
$$d(x, z) \le d(x, y) + d(y, z)$$
Rearranging terms yields:
$$d(x, z) - d(y, z) \le d(x, y) \quad \text{--- (Equation 1)}$$
Similarly, applying the triangle inequality starting with $d(y, z)$:
$$d(y, z) \le d(y, x) + d(x, z)$$
By symmetry of metric $d(y, x) = d(x, y)$, we rewrite:
$$d(y, z) - d(x, z) \le d(x, y) \implies -(d(x, z) - d(y, z)) \le d(x, y) \quad \text{--- (Equation 2)}$$
Combining Equation 1 and Equation 2 using the real absolute value inequality condition (if $a \le c$ and $-a \le c$, then $|a| \le c$):
$$|d(x, z) - d(y, z)| \le d(x, y)$$
Answer: $$|d(x, z) - d(y, z)| \le d(x, y)$$
Question 4: In $\mathbb{R}^2$, verify if $d_\infty(P, Q) = \max(|x_1 - x_2|, |y_1 - y_2|)$ forms a metric, where $P = (x_1, y_1)$ and $Q = (x_2, y_2)$.
Solution:
Let $P=(x_1, y_1), Q=(x_2, y_2), R=(x_3, y_3) \in \mathbb{R}^2$.
1. $d_\infty(P, Q) \ge 0$ because absolute values are non-negative.
2. $d_\infty(P, Q) = 0 \iff \max(|x_1 - x_2|, |y_1 - y_2|) = 0 \iff |x_1 - x_2| = 0$ and $|y_1 - y_2| = 0 \iff x_1 = x_2$ and $y_1 = y_2 \iff P = Q$.
3. $d_\infty(P, Q) = \max(|x_1 - x_2|, |y_1 - y_2|) = \max(|x_2 - x_1|, |y_2 - y_1|) = d_\infty(Q, P)$.
4. For triangle inequality:
$$|x_1 - x_3| \le |x_1 - x_2| + |x_2 - x_3| \le d_\infty(P, Q) + d_\infty(Q, R)$$
$$|y_1 - y_3| \le |y_1 - y_2| + |y_2 - y_3| \le d_\infty(P, Q) + d_\infty(Q, R)$$
Taking the maximum of the left-hand sides:
$$\max(|x_1 - x_3|, |y_1 - y_3|) \le d_\infty(P, Q) + d_\infty(Q, R)$$
$$d_\infty(P, R) \le d_\infty(P, Q) + d_\infty(Q, R)$$
Answer: $$d_\infty \text{ is a valid metric on } \mathbb{R}^2$$
Question 5: Calculate the diameter of the set $A = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 < 4 \}$ equipped with the standard Euclidean metric $d_2$.
Solution:
The set $A$ is an open disk centered at $(0,0)$ with radius $R = \sqrt{4} = 2$.
The diameter of a set $A$ in a metric space is defined as:
$$\text{diam}(A) = \sup \{ d_2(p, q) : p, q \in A \}$$
For any two points $p, q \in A$, by triangle inequality with the origin $O=(0,0)$:
$$d_2(p, q) \le d_2(p, O) + d_2(O, q) < 2 + 2 = 4$$
To show that $4$ is the exact supremum, choose sequence points along the boundary diameter $p_k = \left(-2 + \frac{1}{k}, 0\right)$ and $q_k = \left(2 - \frac{1}{k}, 0\right)$ for $k \ge 1$.
Both $p_k, q_k \in A$ since $\left(\pm\left(2 - \frac{1}{k}\right)\right)^2 < 4$.
The distance between $p_k$ and $q_k$ is:
$$d_2(p_k, q_k) = \sqrt{\left(\left(2 - \frac{1}{k}\right) - \left(-2 + \frac{1}{k}\right)\right)^2 + 0} = 4 - \frac{2}{k}$$
Taking the limit as $k \to \infty$:
$$\lim_{k \to \infty} d_2(p_k, q_k) = 4$$
Since the supremum of distances is $4$, $\text{diam}(A) = 4$.
Answer: $$\text{diam}(A) = 4$$
Question 6: Determine whether the sequence $x_n = \frac{1}{n}$ is a Cauchy sequence in the metric space $((0, 1), d_2)$ where $d_2$ is the standard Euclidean metric, and deduce if $((0, 1), d_2)$ is complete.
Solution:
First, evaluate if $x_n = \frac{1}{n}$ is Cauchy in $(0, 1)$.
For any $\epsilon > 0$, by the Archimedean property, choose $N \in \mathbb{N}$ such that $N > \frac{2}{\epsilon}$.
For all $n, m \ge N$:
$$d_2(x_n, x_m) = \left| \frac{1}{n} - \frac{1}{m} \right| \le \frac{1}{n} + \frac{1}{m} \le \frac{1}{N} + \frac{1}{N} = \frac{2}{N} < \epsilon$$
Hence, $x_n = \frac{1}{n}$ is indeed a Cauchy sequence in $((0, 1), d_2)$.
Next, check completeness: A metric space is complete if every Cauchy sequence in it converges to a point belonging to the space.
In $\mathbb{R}$, $\lim_{n \to \infty} \frac{1}{n} = 0$.
However, $0 \notin (0, 1)$. Therefore, the Cauchy sequence $x_n$ does not converge to any point inside the metric space $((0, 1), d_2)$.
Since there exists a Cauchy sequence that fails to converge within $(0, 1)$, the space is incomplete.
Answer: $$x_n \text{ is Cauchy, but } ((0, 1), d_2) \text{ is NOT complete}$$
Q1: In a discrete metric space $(X, d_D)$, what is the open ball $B(x, 1)$?
Q2: Cantor's Intersection Theorem states that in a complete metric space, every nested sequence of non-empty closed sets $F_1 \supseteq F_2 \supseteq F_3 \dots$ has a non-empty intersection provided that: