Q.Let X = {Ram, Geeta, Akbar} be the set of students of Class XI, who are in school hockey team. Let Y = {Geeta, David, Ashok} be the set of students from Class XI who are in the school football team. Find X ∪ Y and interpret the set

The key idea is Set Union: the set of all elements that belong to at least one of the given sets.

Step 1: List all elements from $X$: Ram, Geeta, Akbar.

Step 2: Add elements from $Y$ that are not already in $X$: David, Ashok. (Geeta is already present.)

Step 3: Combine them into a single set.

$X \cup Y = \{\text{Ram, Geeta, Akbar, David, Ashok}\}$ …

The core idea: What does "union" really mean?

When we take the union of two sets, we are asking: Who is in at least one of these groups? It doesn't matter if a student plays only hockey, only football, or both — if they appear in either set, they belong to the union.

The symbol $\cup$ (cup) is a visual reminder: it opens upward to "collect" everything from both sets. The opposite, $\cap$ (cap), opens downward to "catch" only what's common.

Step-by-step solution

1. Write down the given sets clearly

We have:

• $X = \{\text{Ram}, \text{Geeta}, \text{Akbar}\}$ — hockey team members
• $Y = \{\text{Geeta}, \text{David}, \text{Ashok}\}$ — football team members

2. Identify what the union asks for

$X \cup Y$ means: list every student who is in $X$ or in $Y$ (or both). The "or" here is inclusive — it's the everyday "and/or" of real life.

3. Start listing elements from the first set

Take every name from $X$: Ram, Geeta, Akbar. These are all in the union.

4. Add elements from the second set, skipping repeats

Now take each name from $Y$:

• Geeta — already listed, so skip it.
• David — new, add it.
• Ashok — new, add it.

5. Write the final set

Collecting everything without duplication:

$X \cup Y = \{\text{Ram}, \text{Geeta}, \text{Akbar}, \text{David}, \text{Ashok}\}$ …

Method: Direct Listing (Roster Form)

This is the simplest method for finite sets — we list all distinct elements from both sets without repetition.

Steps

1. Write down the given sets

   $X = \{\text{Ram, Geeta, Akbar}\}$

   $Y = \{\text{Geeta, David, Ashok}\}$

2. Combine all elements from both sets

   Take every name that appears in $X$ or $Y$ (or both).

3. Remove duplicates

   "Geeta" appears in both sets — list it only once.

4. Write the union in roster form

   $X \cup Y = \{\text{Ram, Geeta, Akbar, David, Ashok}\}$

Interpretation

$X \cup Y$ represents the set of all Class XI students who play at least one of the two sports — hockey or football.

• Ram → only hockey …

Let’s break it down.

Step 1: The Correct Answer First

Given:

• $X = \{\text{Ram, Geeta, Akbar}\}$
• $Y = \{\text{Geeta, David, Ashok}\}$

Union $X \cup Y$ = set of all elements that are in $X$ or in $Y$ (or both).

Interpretation: This is the set of all Class XI students who play at least one of the two sports (hockey or football).

Common Mistakes & How to Avoid Them

✗ Mistake 1: Writing a duplicate element (e.g., Geeta twice)

• Wrong: $\{\text{Ram, Geeta, Akbar, Geeta, David, Ashok}\}$
• Why it happens: Students list all elements from $X$ then all from $Y$ without checking for repeats.
• How to avoid: Always merge the sets. A set never has duplicates. After writing the union, scan for any name appearing in both original sets — write it only once.

✗ Mistake 2: Forgetting to include an element (e.g., leaving out Ashok)

• Wrong: $\{\text{Ram, Geeta, Akbar, David}\}$
• Why it happens: Rushing or only listing elements from $X$ and then only the different ones from $Y$, but missing one.
• How to avoid: Use a systematic method:
  1. Write all elements of $X$.
  2. Then add any element from $Y$ that is not already in your list.
  3. Double-check against both original sets.

✗ Mistake 3: Confusing union ($\cup$) with intersection ($\cap$)

• Wrong: $\{\text{Geeta}\}$ (that’s $X \cap Y$)
• Why it happens: Misreading the symbol or mixing up the definitions.
• How to avoid: Remember the keyword:
  • Union ($\cup$) → “or” → take all elements from both.
  • Intersection ($\cap$) → “and” → take only the common ones.
  • A quick trick: $\cup$ looks like a “U” for “Union” = “United” (all together).

✗ Mistake 4: Writing the interpretation incorrectly

• Wrong: “Students who play both hockey and football.”
• Why it happens: Confusing union with intersection again. …