Practice Questions

Consider the set $S = \{1, 2, 3, 4\}$. Create a relation R on S that is reflexive and symmetric but fails to be transitive. Justify your construction.

Explain the difference between a relation and a function with a simple example.

Justify whether the function $f: \mathbf{Z} \to \mathbf{Z}$ defined by $f(x) = |x| + x$ is injective.

Identify if the relation R = {(1,1), (2,2), (3,3)} on set A = {1, 2, 3} is reflexive.

Define a universal relation.

Let the set be $A = \{a, b, c\}$. Analyze if the relation $R = \{(a, a), (b, b), (a, c)\}$ defined on set A is reflexive. Justify your answer.

Describe the conditions for a relation R on a set A to be an equivalence relation.

Analyze if the function $f: \mathbf{Z} \rightarrow \mathbf{Z}$ defined by $f(x) = |x|$ is one-one.

Let $S$ be the set of all straight lines in the Cartesian plane. A relation $R$ is defined on $S$ as $L_1 R L_2$ if $L_1$ is parallel to $L_2$. Justify that $R$ is an equivalence relation. (Note: A line is considered parallel to itself).

Given two functions $f(x) = x^2$ and $g(x) = x + 3$, calculate the value of $(gof)(2)$.

What is a bijective function?

Let R be a relation on the set of integers $\mathbf{Z}$ defined by $aRb$ if $a^2 - b^2$ is divisible by 3. Prove that R is an equivalence relation.

If $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ are given by $f(x) = |x|$ and $g(x) = 5x - 2$, calculate $fog(x)$ and $gof(x)$. Demonstrate that $fog \neq gof$.

Explain what is meant by the domain of a relation R from a set A to a set B.

Summarize the properties of a symmetric relation. Given an example of a relation that is symmetric but not reflexive.

Define an empty relation in a set A.

Let A = {1, 2, 3}. List the elements of the relation R = {(a, b) : |a - b| = 1}. Also, state its domain and range.

Explain why the function f: N → N given by f(x) = x + 1 is one-one but not onto.

Given a function f: {1, 2} → {a, b} defined by f(1) = a, f(2) = b, and a function g: {a, b} → {x, y} defined by g(a) = x, g(b) = y. Describe the composition function gof.

Define reflexive, symmetric, and transitive relations. Provide one example for each type of relation on the set A = {a, b, c}.

Explain the concepts of one-one (injective) and onto (surjective) functions. Give an example of a function that is one-one but not onto, and another example that is onto but not one-one, using finite sets.

Analyze the injectivity and surjectivity of the function $f: \mathbf{R} \rightarrow \mathbf{R}$ given by $f(x) = 3x + 5$.

Let $A = \{1, 2, 3\}$. A relation $R$ is defined on $A$ as $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3)\}$. Examine if $R$ is reflexive, symmetric, or transitive.

Demonstrate that the relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.

Let L be the set of all lines in the XY plane. A relation R is defined on L as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Analyze if R is an equivalence relation.

Create a relation R on the set A = {1, 2, 3} that is symmetric and transitive, but not reflexive. Justify your answer.

Formulate a concise argument to prove that a function $f: A \to B$ from a finite set A to a smaller finite set B (where $|A| > |B|$) cannot be injective.

Create a function $f: \mathbf{N} \to \mathbf{N}$ that is surjective but not injective, and justify both properties.

Let $A = \mathbf{R} - \{2\}$ and $B = \mathbf{R} - \{1\}$. The function $f: A \to B$ is defined by $f(x) = \frac{x-1}{x-2}$. Formulate the inverse function $g: B \to A$ and justify that it is indeed the inverse by showing $g \circ f = I_A$ and $f \circ g = I_B$.

Let $f: A \to B$ and $g: B \to C$ be two surjective functions. Formulate a proof to show that the composition $g \circ f: A \to C$ is also surjective.

Formulate a proof to show that if $R_1$ and $R_2$ are two equivalence relations on a set $A$, then their intersection $R_1 \cap R_2$ is also an equivalence relation on $A$.

Let $f: \mathbf{R} \to \mathbf{R}$ be defined by $f(x) = \sin(x)$ and $g: \mathbf{R} \to \mathbf{R}$ be defined by $g(x) = x^2$. Formulate the composite functions $g \circ f$ and $f \circ g$. Evaluate whether each composite function is injective or surjective, justifying your conclusions.

Let R be a relation on the set of integers $\mathbf{Z}$ defined by $R = \{(a, b) : a + b$ is an even integer$\}.$ Analyze if R is symmetric.

Analyze if the function $f: \mathbf{N} \rightarrow \mathbf{N}$ defined by $f(x) = x + 1$ is onto.

Describe what an equivalence class is. Consider the relation R on the set of integers Z defined by R = {(a, b) : a ≡ b (mod 3)}. Find the equivalence classes [0], [1], and [2].

Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be defined by $f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \ \frac{n+1}{2}, & \text{if } n \text{ is odd} \end{cases}$. Analyze whether the function $f$ is bijective.

Evaluate if the function $f: \mathbf{R} \to \mathbf{R}$ defined by $f(x) = x|x|$ is bijective. Justify your conclusion by checking for injectivity and surjectivity.

Identify the type of relation R in the set Z of integers defined as R = {(x, y): x - y is an even integer}. Explain your choice.

Evaluate the statement: "If the composition $g \circ f$ is one-one, then the function $g$ must be one-one." Justify your conclusion with a counterexample.

Let $\mathbf{N}$ be the set of natural numbers. A relation $R$ is defined on the set $\mathbf{N} \times \mathbf{N}$ by $(a, b) R (c, d)$ if and only if $ad = bc$. Prove that $R$ is an equivalence relation. Furthermore, describe the equivalence class of $(2, 3)$.

Design a function $f: \mathbf{N} \to \mathbf{Z}$ (from the set of natural numbers to the set of integers) that is bijective. Formulate a proof to justify that your designed function is both one-one and onto.

Consider the function $f: \mathbf{R}^+ \rightarrow [4, \infty)$ given by $f(x) = x^2 + 4$, where $\mathbf{R}^+$ is the set of all non-negative real numbers. Demonstrate that $f$ is invertible and calculate its inverse.

Demonstrate that the relation R on the set $\mathbf{N} \times \mathbf{N}$ defined by $(a, b) R (c, d)$ if and only if $a + d = b + c$ is an equivalence relation.

Let $A = \mathbf{R} - \{3\}$ and $B = \mathbf{R} - \{1\}$. Analyze if the function $f: A \rightarrow B$ defined by $f(x) = \frac{x - 2}{x - 3}$ is a bijective function.

Let $f: \mathbf{N} \rightarrow S$ be a function defined as $f(x) = 4x^2 + 12x + 15$, where S is the range of $f$. Demonstrate that $f$ is invertible and calculate the inverse of $f$.