Q.An organization conducted a bike race under 2 different categories — boys and girls. In all, there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets $B$ and $G$ with these participants for his college project. Let $B = \{b_1, b_2, b_3\}$, $G = \{g_1, g_2\}$ where $B$ represents the set of boys selected and $G$ the set of girls who were selected for the final race. Ravi decides to explore these sets for various types of relations and functions. On the basis of the above information, answer the following questions:

Cartesian Product Cardinality: From Intuition to Precision

Imagine ordering a meal: 3 types of bread (roti, naan, paratha) and 4 types of curry (dal, paneer, chicken, fish). How many combinations of one bread and one curry can you make? For each bread you can pair any of the 4 curries, giving $3 \times 4 = 12$ meals. This multiplication is the heart of Cartesian product cardinality.

What is a Cartesian Product?

Given two sets $A$ and $B$, their Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

With $A = \{\text{roti}, \text{naan}, \text{paratha}\}$ and $B = \{\text{dal}, \text{paneer}, \text{chicken}, \text{fish}\}$, $A \times B$ contains 12 pairs like (roti, dal), (naan, paneer), etc.

The Cardinality Statement

The cardinality of the Cartesian product equals the product of the individual set sizes: for each of the $|A|$ choices from $A$, you have $|B|$ choices from $B$, and multiplication counts all pairings.

This generalises to $n$ sets:

A Common Pitfall

For $A = \{1, 2\}$ and $B = \{x, y\}$:

• $A \cup B = \{1, 2, x, y\}$ has size 4
• $A \times B = \{(1,x), (1,y), (2,x), (2,y)\}$ has size $2 \times 2 = 4$ …

One-One Onto (Bijective) Functions

Picture seating students on chairs so that every student gets a chair, every chair is used, no two students share one and none is left empty. A function that manages this perfect pairing between its domain and codomain is one-one onto, or bijective.

One-one (injective)

$f$ is one-one if different inputs always give different outputs — no two students on one chair. Formally, $f(x_1)=f(x_2)\implies x_1=x_2$ (equivalently $x_1\neq x_2\implies f(x_1)\neq f(x_2)$).

$f(x)=2x$ on $\mathbb{R}$ is one-one, since $2a=2b\Rightarrow a=b$. But $f(x)=x^2$ is not: $f(2)=f(-2)=4$ while $2\neq-2$.

Onto (surjective)

$f$ is onto if every element of the codomain is actually hit — no chair left empty. Formally, for every $y$ in the codomain there is some $x$ with $f(x)=y$. Here $f(x)=2x$ is onto (take $x=y/2$), whereas $f:\mathbb{R}\to\mathbb{R},\ f(x)=x^2$ is not, since negative values are never outputs.

Both together — bijective

A function that is one-one and onto is bijective: a one-to-one correspondence in which the two sets match up exactly.

Why it matters

Only a bijection has a genuine inverse function: because each output comes from exactly one input (one-one) and every codomain element is used (onto), the map can be reversed unambiguously. …