Q.Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C

A universal set for $A=\{1,3,5\}$, $B=\{2,4,6\}$, $C=\{0,2,4,6,8\}$ must contain every element of all three, i.e. all of $A\cup B\cup C = \{0,1,2,3,4,5,6,8\}$.

Checking each option: (i) $\{0,...,6\}$ is missing $8$.

(ii) $\phi$ has no elements.

(iii) $\{0,...,10\}$ contains all required elements. …

A universal set $U$ for given sets must satisfy $A \subseteq U$, $B \subseteq U$, and $C \subseteq U$ simultaneously -- every element that appears anywhere in $A$, $B$, or $C$ must also appear in $U$.

Step 1: Find the elements $U$ must contain.

$A = \{1,3,5\}$, $B = \{2,4,6\}$, $C = \{0,2,4,6,8\}$.

Any valid universal set must contain at least these eight elements.

Step 2: Check each option.

1. $\{0,1,2,3,4,5,6\}$ -- missing $8$ (which is in $C$). Not valid.
2. $\phi$ (empty set) -- contains no elements at all. Not valid.
3. $\{0,1,2,3,4,5,6,7,8,9,10\}$ -- contains $0,1,2,3,4,5,6,8$ and more. Valid. …

Method: Universal Set Verification by Superset Condition

Concept (Why this works)

A universal set ($U$) for given sets must contain every element from all those sets.

If even one element from $A$, $B$, or $C$ is missing from $U$, then $U$ cannot be a universal set for them.

Steps

Step 1: List all distinct elements in $A$, $B$, and $C$

• $A = \{1, 3, 5\}$
• $B = \{2, 4, 6\}$
• $C = \{0, 2, 4, 6, 8\}$

Combine: $\{0, 1, 2, 3, 4, 5, 6, 8\}$

Step 2: For each candidate set, check if it contains every element from Step 1

• (i) $\{0, 1, 2, 3, 4, 5, 6\}$

  Missing: $8$ (from $C$) → No

• (ii) $\phi$ (empty set)

  Missing: all elements → No

• (iii) $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

  Contains: $0,1,2,3,4,5,6,8$ → Yes …

Common Mistakes & How to Avoid Them

Mistake 1: Thinking the Universal Set Must Be Exactly the Union

The error: Students often believe the universal set must be precisely $A \cup B \cup C = \{0,1,2,3,4,5,6,8\}$ and reject any set that contains extra elements.

Why it's wrong: A universal set is any set that contains all elements of the given sets. It can have extra elements — that's perfectly fine.

How to avoid: Remember the definition: $U$ is a universal set for $A, B, C$ if and only if $A \subseteq U$, $B \subseteq U$, and $C \subseteq U$. Extra elements don't break this condition.

Mistake 2: Accepting the Empty Set as a Universal Set

The error: Some students think $\phi$ (empty set) could work because "it contains nothing, so it contains all sets."

Why it's wrong: The empty set contains no elements. For $A = \{1,3,5\}$ to be a subset of $\phi$, every element of $A$ must be in $\phi$ — but $1 \notin \phi$, $3 \notin \phi$, $5 \notin \phi$.

How to avoid: Test with one element from each given set. If even a single element is missing from the candidate universal set, reject it immediately.

Mistake 3: Forgetting to Check All Three Sets

The error: Students check only one or two sets and assume the candidate works.

Example: Option (i) $\{0,1,2,3,4,5,6\}$ contains $A$ and $B$, but does it contain $C = \{0,2,4,6,8\}$? No — $8 \notin$ option (i).

How to avoid: Make a checklist:

• Does it contain every element of $A$? ✓ / ✗
• Does it contain every element of $B$? ✓ / ✗
• Does it contain every element of $C$? ✓ / ✗

Only accept if all three are ✓.

Mistake 4: Confusing "May Be" with "Must Be"

The error: The question asks "which may be considered" — meaning multiple answers are possible. Students sometimes pick only one.

Why it's wrong: Options (iii) and (iv) both work:

• (iii) $\{0,1,2,3,4,5,6,7,8,9,10\}$ contains all elements of $A, B, C$ ✓
• (iv) $\{1,2,3,4,5,6,7,8\}$ contains all elements of $A, B, C$ ✓

How to avoid: Read the question carefully — "which of the following may be" signals multiple correct answers are possible.

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