Q.Let the relation $R$ be defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(a, b) : |a^2 - b^2| < 8\}$. Then $R$ is given by ______.

Concept: Relation Properties — we first list all ordered pairs that satisfy the condition, then check which pairs are present.

Step 1: For each $a, b \in A$, compute $|a^2 - b^2|$ and check if it is $< 8$.

Step 2:

• $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$. Differences less than 8 occur only between numbers whose squares differ by at most 7.

Step 3: List all pairs:

• With $a=1$: $|1-1|=0$, $|1-4|=3$, $|1-9|=8$ (not included) → $(1,1), (1,2)$ and symmetrically $(2,1)$.
• With $a=2$: $|4-1|=3$, $|4-4|=0$, $|4-9|=5$, $|4-16|=12$ → $(2,1), (2,2), (2,3)$ and $(3,2)$.
• With $a=3$: $|9-4|=5$, $|9-9|=0$, $|9-16|=7$, $|9-25|=16$ → $(3,2), (3,3), (3,4)$ and $(4,3)$. …

We need all ordered pairs $(a,b)$ with $a,b\in\{1,2,3,4,5\}$ satisfying $|a^2-b^2|<8$. The squares are $1,4,9,16,25$.

• Every pair with $a=b$ gives $|a^2-b^2|=0<8$: $(1,1),(2,2),(3,3),(4,4),(5,5)$.
• $(1,2)$ and $(2,1)$: $|1-4|=3<8$ ✓
• $(2,3)$ and $(3,2)$: $|4-9|=5<8$ ✓
• $(3,4)$ and $(4,3)$: $|9-16|=7<8$ ✓ …

Method: Building a Relation in Roster Form From a Condition

Use this when a relation on a small finite set is defined by a condition such as $|a^2-b^2|<8$ and you must list its pairs.

Steps

Step 1: Precompute the key quantity for each element.

Here, list the squares $a^2$ for every $a$ in the set so comparisons are quick.

Step 2: Test each ordered pair against the condition. …

Common Mistakes

Mistake 1: Including a pair that meets the boundary exactly.

Why it's wrong: the condition is $|a^2-b^2|<8$ (strict), so a pair like $(1,3)$ with $|1-9|=8$ does not qualify. Correct approach: admit only differences strictly less than $8$.

Mistake 2: Listing $(a,b)$ but forgetting $(b,a)$. …