Q.In the following, state whether A = B or not:

Concept: Set Equality

Two sets are equal if and only if they contain exactly the same elements. Order and repetition do not matter—only membership.

(i) $A = \{a, b, c, d\}$ and $B = \{d, c, b, a\}$

Both sets contain precisely the elements $a, b, c, d$. The different listing order is irrelevant.

$A = B$

(ii) $A = \{4, 8, 12, 16\}$ and $B = \{8, 4, 16, 18\}$

$A$ contains $12$ but $B$ does not; $B$ contains $18$ but $A$ does not.

$A \neq B$

(iii) $A = \{2, 4, 6, 8, 10\}$ and $B = \{x : x \text{ is a positive even integer and } x \leq 10\}$

The positive even integers up to $10$ are exactly $2, 4, 6, 8, 10$, so $B = \{2, 4, 6, 8, 10\}$.

$A = B$ …

Understanding Set Equality

The fundamental principle of set equality is beautifully simple: two sets are equal when every element of one is an element of the other, and vice versa. Order doesn't matter—$\{1, 2, 3\}$ and $\{3, 1, 2\}$ are the same set. Repetition doesn't matter either—$\{1, 2, 2, 3\}$ is just $\{1, 2, 3\}$. What matters is membership: does each set contain precisely the same collection of distinct objects?

This is the membership test: $A = B$ if and only if $(x \in A \Rightarrow x \in B)$ and $(x \in B \Rightarrow x \in A)$ for all $x$.

Let me examine each pair systematically.

(i) $A = \{ a, b, c, d \}$ and $B = \{ d, c, b, a \}$

1. List the elements of $A$: $a, b, c, d$

2. List the elements of $B$: $d, c, b, a$

3. Check if every element of $A$ is in $B$:

   • Is $a \in B$? Yes.
   • Is $b \in B$? Yes.
   • Is $c \in B$? Yes.
   • Is $d \in B$? Yes.

4. Check if every element of $B$ is in $A$:

   • Is $d \in A$? Yes.
   • Is $c \in A$? Yes.
   • Is $b \in A$? Yes.
   • Is $a \in A$? Yes.

Both sets contain exactly the same four elements. The different ordering is irrelevant.

Conclusion: $A = B$ ✓

(ii) $A = \{ 4, 8, 12, 16 \}$ and $B = \{ 8, 4, 16, 18\}$

1. List the elements of $A$: $4, 8, 12, 16$

2. List the elements of $B$: $8, 4, 16, 18$

3. Check if every element of $A$ is in $B$:

   • Is $4 \in B$? Yes.
   • Is $8 \in B$? Yes.
   • Is $12 \in B$? No. $12$ is not in $B$.

We can stop here. Since $12 \in A$ but $12 \notin B$, the sets are not equal.

Alternatively, notice that $18 \in B$ but $18 \notin A$, which also proves inequality.

Conclusion: $A \neq B$ ✗

(iii) $A = \{2, 4, 6, 8, 10\}$ and $B = \{ x : x \text{ is positive even integer and } x \leq 10\}$

1. Understand set $B$ by listing its elements. The positive even integers up to and including 10 are: $2, 4, 6, 8, 10$.

   So $B = \{2, 4, 6, 8, 10\}$ in roster form.

2. Compare with $A$: $A = \{2, 4, 6, 8, 10\}$

3. Both sets contain exactly the same five elements: $2, 4, 6, 8, 10$.

Conclusion: $A = B$ ✓ …

Method: Set Equality by Element Comparison

Two sets are equal if and only if every element of A is in B and every element of B is in A. Order and repetition do not matter.

Steps:

1. List all distinct elements of set A.
2. List all distinct elements of set B.
3. Check: Is each element of A present in B?
4. Check: Is each element of B present in A?
5. If both checks pass → A = B. Otherwise → A ≠ B.

(i) A = { a, b, c, d }, B = { d, c, b, a }

• Elements of A: a, b, c, d
• Elements of B: d, c, b, a
• Every element of A is in B ✓
• Every element of B is in A ✓

Answer: A = B ✓

(ii) A = { 4, 8, 12, 16 }, B = { 8, 4, 16, 18 }

• Elements of A: 4, 8, 12, 16
• Elements of B: 8, 4, 16, 18
• 12 is in A but not in B ✗

Answer: A ≠ B ✗

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Common Mistakes Students Make on Set Membership & Equality

Here are the most frequent errors students commit when comparing sets, with clear explanations and avoidance strategies.

Mistake 1: Confusing Order with Equality

The Error: Students think that because elements are listed in a different order, the sets are different.

Example from the question: In (i), $A = \{a, b, c, d\}$ and $B = \{d, c, b, a\}$.

Why it's wrong: Sets are unordered collections. The order in which elements are written has no meaning. Two sets are equal if and only if they contain exactly the same elements.

How to avoid: Always mentally rearrange the elements of both sets in the same order (alphabetical or numerical) before comparing. Ask yourself: "Does every element of A appear in B, and does every element of B appear in A?"

Correct answer for (i): $A = B$ ✓

Mistake 2: Ignoring One Wrong Element

The Error: Students see that most elements match and declare the sets equal, overlooking a single mismatched element.

Example from the question: In (ii), $A = \{4, 8, 12, 16\}$ and $B = \{8, 4, 16, 18\}$.

Why it's wrong: $A$ contains $12$, but $B$ does not. $B$ contains $18$, but $A$ does not. Even one mismatched element makes the sets unequal.

How to avoid: Use the two-way check:

• Is every element of $A$ in $B$? (Check: 4✓, 8✓, 12✗, 16✓ → No)
• Is every element of $B$ in $A$? (Check: 8✓, 4✓, 16✓, 18✗ → No)

Correct answer for (ii): $A \neq B$ ✗

Mistake 3: Misreading Set-Builder Notation

The Error: Students fail to correctly interpret the condition in set-builder form $\{x : \text{condition}\}$.

Example from the question: In (iii), $B = \{x : x \text{ is positive even integer and } x \leq 10\}$.

Why it's wrong: Students might forget that "positive even integer" means $2, 4, 6, 8, 10, \dots$ and the condition $x \leq 10$ limits it to $\{2, 4, 6, 8, 10\}$.

How to avoid: Always list out the elements of the set-builder form explicitly before comparing:

• $B = \{2, 4, 6, 8, 10\}$
• Now compare with $A = \{2, 4, 6, 8, 10\}$

Correct answer for (iii): $A = B$ ✓

Mistake 4: Assuming Patterns Continue Indefinitely …