Relations

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is any subset of the Cartesian product $A \times B$.

That is, $R \subseteq A \times B$.

The subset is formed by picking only those ordered pairs $(x, y)$ from $A \times B$ for which a specific condition or relationship holds between $x$ (the first element) and $y$ (the second element).

The second element $y$ in such an ordered pair is called the image of the first element $x$.

Intuition: A relation is like a filter on the full list of all possible pairings — you keep only those pairs where the two things are connected in a particular way.

Example:

Let $A = \{1, 2\}$ and $B = \{3, 4\}$.

Then $A \times B = \{(1,3), (1,4), (2,3), (2,4)\}$.

Define a relation $R$ by the rule "$y$ is one more than $x$" (i.e. $y = x + 1$). …

The domain of a relation $R$ from a set $A$ to a set $B$ is the set of all first elements of the ordered pairs that belong to $R$.

In other words, if $R \subseteq A \times B$ and you list every ordered pair $(x, y)$ in $R$, then the domain is the collection of all the $x$'s that actually appear in those pairs. The textbook (Definition 3) states this exactly: "The set of all first elements of the ordered pairs in a relation $R$ from a set $A$ to a set $B$ is called the domain of the relation $R$."

Intuition: Think of a relation as a set of connections — each connection is an arrow from some element of $A$ to some element of $B$. The domain is simply the group of "starting points" from $A$ that actually have at least one arrow leaving them. Not every element of $A$ needs to be in the domain; only those that are used as a first coordinate in some ordered pair of $R$. …

Definition 4 states: The set of all second elements in a relation $R$ from a set $A$ to a set $B$ is called the range of the relation $R$. The whole set $B$ is called the codomain of the relation $R$. Note that $\text{range} \subset \text{codomain}$.

In other words, if $R \subseteq A \times B$ is a relation, then:

• Range of $R$ = $\{ y \in B \mid (x, y) \in R \text{ for some } x \in A \}$.
• Codomain of $R$ = the entire set $B$ (given in advance, regardless of which elements actually appear in $R$).

The range is always a subset of the codomain — it collects only those second elements that actually get paired with some first element. The codomain is the full target set, which may contain elements that never appear as a second element in any ordered pair of $R$.

Intuition: Think of a relation as a matching between two groups. The codomain is the whole group on the right side; the range is just the set of people from that group who actually get matched.

Concrete example (from Example 7 of the textbook):

Let $A = \{1, 2, 3, 4, 5, 6\}$ and define $R = \{(x, y) : y = x + 1\}$. Then:

• $R = \{(1,2), (2,3), (3,4), (4,5), (5,6)\}$. …

What Fig. 2.4 Shows

The arrow diagram presents two sets side by side. On the left is set $P = \{a, b, c\}$, written inside an oval. On the right is set $Q = \{\text{Ali, Bhanu, Binoy, Chandra, Divya}\}$, inside another oval. Indigo arrows connect elements of $P$ to elements of $Q$ according to a specific rule: $a$ points to Ali, $b$ points to both Bhanu and Binoy, and $c$ points to Chandra. Divya receives no arrow at all.

This is not just a random collection of arrows. The diagram is a visual representation of a relation — a subset of the Cartesian product $P \times Q$. The rule that decides which arrows exist is: $x$ is the first letter of the name $y$. So $a$ is the first letter of Ali, $b$ is the first letter of both Bhanu and Binoy, and $c$ is the first letter of Chandra. Divya starts with D, which is not in $P$, so no arrow reaches her.

The Physical Idea

The figure teaches that a relation is fundamentally a matching rule between two sets. The Cartesian product $P \times Q$ contains all 15 possible ordered pairs, but the relation picks out only those pairs that satisfy the given condition. The arrow diagram shows this selection at a glance: each arrow corresponds to one ordered pair in the relation.

The textbook then formalises this into three definitions. The domain is the set of all first elements that actually appear in the relation — here $\{a, b, c\}$, which happens to be the whole of $P$. The range is the set of all second elements that receive arrows — $\{\text{Ali, Bhanu, Binoy, Chandra}\}$. The codomain is the entire set $Q$, which includes Divya even though she is not related to any element of $P$.

The Key Formula

The central result that this figure introduces is the number of possible relations between two finite sets. If $n(A) = p$ and $n(B) = q$, then:

Why $2^{pq}$? A relation is a subset of $A \times B$. The Cartesian product has $pq$ ordered pairs. For each pair, you have two choices: include it in the relation or leave it out. So the total number of distinct relations equals the number of subsets of a set with $pq$ elements, which is $2^{pq}$. …

Fig 2.5 is the arrow diagram for the relation $R = \{(x, y) : y = x + 1\}$ defined on the set $A = \{1, 2, 3, 4, 5, 6\}$. The diagram uses two vertical lists (often drawn as ovals or columns) of the numbers 1 through 6, placed side by side. The left list represents the domain (the set of first elements $x$), and the right list represents the codomain (the set of possible second elements $y$, which is also $A$ here).

From each element in the left list, an arrow slants downward to the right, pointing to the element in the right list that is exactly one more. So you see arrows: $1 \to 2$, $2 \to 3$, $3 \to 4$, $4 \to 5$, $5 \to 6$. The element $6$ on the left has no arrow leaving it — because $6 + 1 = 7$, and $7$ is not in the set $A$. This visual gap is the whole point of the figure: it shows that a relation need not pair every element of the domain with an image.

The relation itself is given by the rule $y = x + 1$, which generates the ordered pairs $(1,2), (2,3), (3,4), (4,5), (5,6)$. The arrow diagram makes it instantly clear that the domain is $\{1, 2, 3, 4, 5\}$ (the elements that actually have arrows), the range is $\{2, 3, 4, 5, 6\}$ (the elements that receive arrows), and the codomain is the full set $\{1, 2, 3, 4, 5, 6\}$.

The central formula the textbook develops with this figure is simply the rule defining the relation:

Fig 2.6 is an arrow diagram that shows a relation between two sets, P and Q. The set P is written on the left as a slate oval containing the numbers 9, 4, and 25. The set Q is on the right as another slate oval containing the numbers 5, 3, 2, 1, −2, −3, and −5. Crossing indigo arrows connect elements of P to elements of Q: 9 points to both 3 and −3, 4 points to both 2 and −2, and 25 points to both 5 and −5. The element 1 in Q has no arrow coming to it from any element of P.

The relation shown is “x is the square of y”. In set-builder form, this is written as

In roster form, the relation is

The domain of this relation is the set of all first elements: $\{4, 9, 25\}$. The range is the set of all second elements that actually appear: $\{-2, 2, -3, 3, -5, 5\}$. The codomain is the whole set Q, which includes 1 as well — but 1 has no pre-image in P, so it is not part of the range.

Fig 2.7 is the arrow diagram for the relation defined by $y = x - 2$ between the sets $P = \{5,6,7\}$ and $Q = \{3,4,5\}$. The diagram shows two ovals: the left oval labelled $P$ contains the numbers 5, 6, 7; the right oval labelled $Q$ contains 3, 4, 5. Three horizontal arrows run from left to right, each connecting an element of $P$ to its image in $Q$: $5 \to 3$, $6 \to 4$, and $7 \to 5$.

The physical idea is straightforward: each element $x$ in $P$ is related to exactly one element $y$ in $Q$ through the rule "subtract 2". This is a relation — a subset of the Cartesian product $P \times Q$. In roster form, the relation is

In set-builder form, it is

From the diagram you can read off the three key sets:

• Domain = $\{5,6,7\}$ (all first elements, i.e., the entire set $P$).
• Range = $\{3,4,5\}$ (all second elements that actually appear; here it equals $Q$).
• Codomain = $\{3,4,5\}$ (the whole set $Q$).