1.
If $A = \{1, 2\}$ and $B = \{a, b\}$, write down the Cartesian product $A \times B$.
2.
If $n(A) = 3$ and $n(B) = 4$, what is the total number of elements in $A \times B$?
3.
True or False: $A \times B$ is always equal to $B \times A$.
4.
If the ordered pairs $(x+1, y-2)$ and $(3, 1)$ are equal, find the values of $x$ and $y$.
5.
Let $A = \{1, 2, 3\}$. Write the subset of $A \times A$ where the first element is strictly greater than the second element.
6.
What is a relation $R$ from a non-empty set $A$ to a non-empty set $B$ defined as in terms of sets?
7.
If $R = \{(1,2), (2,3), (3,4)\}$, write the domain and range of the relation $R$.
8.
If $A = \{a, b, c\}$ and $B = \{1, 2\}$, what is the total number of possible relations from $A$ to $B$?
9.
Is an empty set $\phi$ considered a valid relation on a set $A$? (Yes/No)
10.
What do we call a relation on set $A$ where every element of $A$ is related to every element of $A$?
11.
If $A = \{1, 2, 3\}$, write out the identity relation $I_A$ in roster form.
12.
Fill in the blank: A relation $R$ on set $A$ is called reflexive if $(a, a) \in R$ for all $a \in$ _______.
13.
Let $A = \{1, 2, 3\}$ and $R = \{(1,2), (2,1)\}$. Is the relation $R$ symmetric?
14.
Let $A = \{1, 2, 3\}$ and $R = \{(1,1), (2,2), (3,3), (1,2)\}$. Is the relation $R$ reflexive?
15.
True or False: If a relation is symmetric and transitive, it must always be reflexive.
16.
What three properties must a relation possess to be called an Equivalence Relation?
17.
Let $R$ be a relation on integers $\mathbb{Z}$ defined by $a R b$ if $a - b$ is an even number. Is $R$ reflexive?
18.
In the relation from Question 17, if $a R b$, does it imply that $b R a$?
19.
For an equivalence relation on set $A$, what do we call the subset of elements in $A$ that are all related to a specific element $x$?
20.
Find the equivalence class of $0$, denoted by $[0]$, for the relation $a R b \implies a-b$ is a multiple of $2$ on $\mathbb{Z}$.
21.
What is the essential condition for a relation from set $A$ to set $B$ to be considered a function?
22.
In a function $f: A \to B$, what is the set $B$ formally called?
23.
What is the "Horizontal Line Test" used to determine when looking at the graph of a function?
24.
Fill in the blank: A function $f: A \to B$ is one-one (injective) if $f(x_1) = f(x_2) \implies$ _______.
25.
If the range of a function $f$ is exactly equal to its co-domain, the function is called _______.
26.
What do we call a function that is both injective (one-one) and surjective (onto)?
27.
Classify $f(x) = x^2$ mapping from $\mathbb{R} \to \mathbb{R}$ as one-one or many-one.
28.
Classify $f(x) = 2x + 3$ mapping from $\mathbb{R} \to \mathbb{R}$ as one-one or many-one.
29.
Is $f(x) = \sin(x)$ mapping from $\mathbb{R} \to \mathbb{R}$ an onto function?
30.
If there is at least one element in the co-domain of a function that does not have a pre-image in the domain, what type of mapping is it?
31.
If $n(A) = p$ and $n(B) = q$, what is the formula for the total number of functions from $A$ to $B$?
32.
If $n(A) = 3$ and $n(B) = 3$, find the total number of bijective functions from $A$ to $B$.
33.
If set $A$ has 4 elements and set $B$ has 2 elements, how many one-one functions can be defined from $A$ to $B$?
34.
If $f(x) = x^2$ and $g(x) = 2x$, find the value of $(f \circ g)(x)$.
35.
Using the functions from Question 34, find the value of $(g \circ f)(x)$.
36.
True or False: The composition of functions is always commutative i.e., $f \circ g = g \circ f$.
37.
If $f: A \to B$ and $g: B \to C$, then the composite function $g \circ f$ maps from set _______ to set _______.
38.
If $f(x) = x + 1$, find the expression for $f(f(x))$.
39.
A function $f: A \to B$ is invertible if and only if it is a _______ function.
40.
If a bijective function is given by $f(x) = 2x + 3$, find its inverse $f^{-1}(y)$.
41.
True or False: The inverse of a bijective function is also a bijective function.
42.
State the reversal rule for the inverse of composite functions: $(g \circ f)^{-1} = $ _______.
43.
A binary operation $*$ on set $A$ is a function from mapping set _______ to set _______.
44.
An operation $*$ is considered commutative on set $A$ if $a * b = $ ______ for all $a, b \in A$.
45.
An element $e \in A$ is the identity element for $*$ if $a * e = e * a = $ ______.
46.
Let $*$ be a binary operation defined on $\mathbb{Z}$ by $a * b = a + b + 1$. Calculate the value of $2 * 3$.
47.
Is the operation defined in Question 46 commutative? (Yes/No)
48.
In addition modulo $5$ (denoted $+_5$), find the value of $3 +_5 4$.
49.
In multiplication modulo $6$ (denoted $\times_6$), find the value of $4 \times_6 5$.
50.
Let $*$ be a binary operation defined on Natural Numbers $\mathbb{N}$ by $a * b = \text{max}(a, b)$. Evaluate $5 * 7$.
