Q.Find the union of each of the following pairs of sets :

Concept: Subset Listing — The union of two sets is the set of all elements that belong to at least one of the sets.

(i) $X = \{1, 3, 5\}$, $Y = \{1, 2, 3\}$

List all distinct elements from both: $1, 2, 3, 5$.

$X \cup Y = \{1, 2, 3, 5\}$

(ii) $A = \{a, e, i, o, u\}$, $B = \{a, b, c\}$

Distinct elements: $a, b, c, e, i, o, u$.

$A \cup B = \{a, b, c, e, i, o, u\}$

(iii) $A = \{3, 6, 9, 12, \dots\}$, $B = \{1, 2, 3, 4, 5\}$

Common element: $3$. Union includes all of $B$ plus multiples of $3$ beyond $5$.

$A \cup B = \{1, 2, 3, 4, 5, 6, 9, 12, \dots\}$

Or: $\{x : x \in \mathbb{N}, x < 6 \text{ or } x \text{ is a multiple of } 3\}$

(iv) $A = \{2, 3, 4, 5, 6\}$, $B = \{7, 8, 9\}$

No overlap.

$A \cup B = \{2, 3, 4, 5, 6, 7, 8, 9\}$

(v) $A = \{1, 2, 3\}$, $B = \phi$

Union with empty set gives the other set.

$A \cup B = \{1, 2, 3\}$

The union of two sets $A$ and $B$, written $A \cup B$, is the set of all elements that are in $A$, or in $B$, or in both. There is no duplication — if an element appears in both sets, it appears only once in the union.

The simplest way to find $A \cup B$ is to list every element from $A$, then add any element from $B$ that hasn't already been listed. That is exactly what we will do for each pair.

(i) $X = \{1, 3, 5\}$, $Y = \{1, 2, 3\}$

1. Start with all elements of $X$: $\{1, 3, 5\}$.
2. Now look at $Y$: it has $1$, $2$, and $3$. The $1$ and $3$ are already in our list, so we only add the new element $2$.
3. The combined list is $\{1, 2, 3, 5\}$.

(ii) $A = \{a, e, i, o, u\}$, $B = \{a, b, c\}$

1. Start with $A$: $\{a, e, i, o, u\}$.
2. From $B$, $a$ is already present; add $b$ and $c$.
3. Result: $\{a, b, c, e, i, o, u\}$.

(iii) $A = \{x : x \text{ is a natural number and a multiple of } 3\}$, $B = \{x : x \text{ is a natural number less than } 6\}$

First, write both sets explicitly.

• $A = \{3, 6, 9, 12, 15, \dots\}$ — all positive multiples of $3$.
• $B = \{1, 2, 3, 4, 5\}$ — natural numbers $1$ through $5$.

1. Start with $A$: $\{3, 6, 9, 12, \dots\}$.
2. From $B$, the element $3$ is already in $A$; add $1, 2, 4, 5$.
3. The union is $\{1, 2, 3, 4, 5, 6, 9, 12, 15, \dots\}$.

(iv) $A = \{x : x \text{ is a natural number and } 1 < x \leq 6\}$, $B = \{x : x \text{ is a natural number and } 6 < x < 10\}$

Write them out:

• $A = \{2, 3, 4, 5, 6\}$ (natural numbers greater than $1$ and up to $6$).
• $B = \{7, 8, 9\}$ (natural numbers strictly between $6$ and $10$).

1. Start with $A$: $\{2, 3, 4, 5, 6\}$.
2. $B$ has $7, 8, 9$ — none of these are in $A$, so add all three.
3. Result: $\{2, 3, 4, 5, 6, 7, 8, 9\}$.

Notice that $6$ is in $A$ but not in $B$, and $7$ is in $B$ but not in $A$. The sets are disjoint (no common elements), so the union is simply the combination of both.

(v) $A = \{1, 2, 3\}$, $B = \phi$ (the empty set)

1. Start with $A$: $\{1, 2, 3\}$.
2. $B$ has no elements, so nothing to add.
3. The union is exactly $A$: $\{1, 2, 3\}$.

Method: Roster Listing & Direct Definition

This method works by first writing each set in roster form (listing all elements explicitly), then combining all distinct elements into one set — that is the union.

Steps:

1. Write each set in roster form (list all elements inside curly braces).
2. Combine all elements from both sets into a single list.
3. Remove duplicates — each element appears only once.
4. Write the final set in roster form.

(i) $X = \{1, 3, 5\}$, $Y = \{1, 2, 3\}$

• Already in roster form.
• Combine: $1, 3, 5, 1, 2, 3$
• Remove duplicates: $\{1, 2, 3, 5\}$

Answer:

(ii) $A = \{a, e, i, o, u\}$, $B = \{a, b, c\}$

• Already in roster form.
• Combine: $a, e, i, o, u, a, b, c$
• Remove duplicates: $\{a, b, c, e, i, o, u\}$

Answer:

(iii) $A = \{x : x \text{ is a natural number and multiple of 3}\}$, $B = \{x : x \text{ is a natural number less than 6}\}$

• Step 1 — Roster form:

  $A = \{3, 6, 9, 12, \dots\}$ (infinite)

  $B = \{1, 2, 3, 4, 5\}$

• Step 2 — Combine: All multiples of 3 together with $1, 2, 3, 4, 5$.

• Step 3 — Remove duplicates: $3$ appears in both, keep once.

• Step 4 — Write union:

Note: Since $A$ is infinite, the union is also infinite — we describe it as "all natural numbers that are either less than 6 or multiples of 3."

(iv) $A = \{x : x \text{ is a natural number and } 1 < x \leq 6\}$, $B = \{x : x \text{ is a natural number and } 6 < x < 10\}$

• Roster form:

  $A = \{2, 3, 4, 5, 6\}$

  $B = \{7, 8, 9\}$

• Combine: $2, 3, 4, 5, 6, 7, 8, 9$ — no duplicates.

• Final:

(v) $A = \{1, 2, 3\}$, $B = \phi$ (empty set)

• Roster form: $A = \{1, 2, 3\}$, $B = \{\}$
• Combine: Only elements from $A$ exist.
• Final:

Key idea: The union with an empty set always gives the other set itself.

(i) X = {1, 3, 5}, Y = {1, 2, 3}

✓ Correct answer:

$X \cup Y = \{1, 2, 3, 5\}$

✗ Common mistakes:

• Writing $\{1, 2, 3\}$ (forgetting 5).
• Writing $\{1, 1, 2, 3, 3, 5\}$ (repeating elements).

🛡️ How to avoid:

• Union means all elements from both sets — no element is left out.
• Never repeat elements in set notation. List each element only once.

(ii) A = {a, e, i, o, u}, B = {a, b, c}

✓ Correct answer:

$A \cup B = \{a, b, c, e, i, o, u\}$

✗ Common mistakes:

• Writing $\{a, b, c, e, i, o, u\}$ in wrong order — order doesn't matter, but some students think it does.
• Forgetting vowels like 'e', 'i', 'o', 'u' — only writing $\{a, b, c\}$.

🛡️ How to avoid:

• Order is irrelevant in sets. Just list all distinct letters.
• Check both sets carefully — don't assume one set is "more important".

(iii) A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}

✓ Correct answer:

• $A = \{3, 6, 9, 12, \dots\}$
• $B = \{1, 2, 3, 4, 5\}$
• $A \cup B = \{1, 2, 3, 4, 5, 6, 9, 12, \dots\}$

✗ Common mistakes:

• Writing only $\{1, 2, 3, 4, 5, 6\}$ — forgetting that A is infinite.
• Writing $\{3, 6, 9, \dots\}$ — forgetting B entirely.

🛡️ How to avoid:

• Identify infinite sets — if a set has "...", the union is also infinite.
• Write the finite part first, then the infinite pattern: $\{1, 2, 3, 4, 5, 6, 9, 12, \dots\}$.

(iv) A = {x : x is a natural number and $1 < x \leq 6$}, B = {x : x is a natural number and $6 < x < 10$}

✓ Correct answer:

• $A = \{2, 3, 4, 5, 6\}$
• $B = \{7, 8, 9\}$
• $A \cup B = \{2, 3, 4, 5, 6, 7, 8, 9\}$

✗ Common mistakes:

• Including 1 in A (forgetting $1 < x$, not $1 \leq x$).
• Including 6 in B (forgetting $6 < x$, not $6 \leq x$).
• Including 10 in B (forgetting $x < 10$).

🛡️ How to avoid:

• Draw a number line for inequalities — mark open/closed circles.
• Check endpoints carefully: $<$ means open, $\leq$ means closed.

(v) A = {1, 2, 3}, B = $\phi$

✓ Correct answer:

$A \cup \phi = \{1, 2, 3\}$

✗ Common mistakes:

• Writing $\phi$ as the answer (thinking union with empty set gives empty set).
• Writing $\{ \phi \}$ (treating empty set as an element).

🛡️ How to avoid:

• Remember the rule: $X \cup \phi = X$ for any set $X$.
• Empty set has no elements — union adds nothing.

📌 Quick Summary Table

Final tip: Always write the sets explicitly before finding the union — especially for set-builder forms. This single step prevents most errors.