Q.Are the following pair of sets equal ? Give reasons.

Concept: Set Membership — two sets are equal if and only if they contain exactly the same elements.

(i) Solve $x^2 + 5x + 6 = 0$.

Factorising: $(x+2)(x+3)=0$, so $x = -2, -3$.

Thus $B = \{-2, -3\}$, while $A = \{2, 3\}$. …

(i) $A = \{2, 3\}$, $B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}$

Concept first: Set equality means every element of $A$ is in $B$ and every element of $B$ is in $A$. So we need to find the actual elements of $B$ by solving the quadratic.

1. Solve the equation.

   $x^2 + 5x + 6 = 0$ factors as $(x + 2)(x + 3) = 0$.

   So the solutions are $x = -2$ and $x = -3$.

   Therefore $B = \{-2, -3\}$.

2. Compare the sets.

   $A = \{2, 3\}$ contains positive numbers; $B = \{-2, -3\}$ contains negative numbers.

   No element of $A$ is in $B$, and vice versa.

3. Conclusion: $A \neq B$.

(ii) $A = \{x : x \text{ is a letter in the word FOLLOW}\}$, $B = \{y : y \text{ is a letter in the word WOLF}\}$

Concept first: A set lists each distinct element only once. Repeated letters in a word don't create repeated elements in the set.

1. List the letters in FOLLOW. The word is F-O-L-L-O-W. The distinct letters are F, O, L, W. …

Method: Set Equality via Element Listing

Two sets are equal if and only if they contain exactly the same elements — no more, no less. The method is:

1. List all elements of each set explicitly (in roster form).
2. Compare the elements — check that every element of A is in B and every element of B is in A.
3. Conclude equality or inequality with a reason.

(i) $A = \{2, 3\}$, $B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}$

Step 1: List elements of B

Solve $x^2 + 5x + 6 = 0$

Factor: $(x + 2)(x + 3) = 0$

So $x = -2$ or $x = -3$

Thus $B = \{-2, -3\}$

Step 2: Compare

$A = \{2, 3\}$ and $B = \{-2, -3\}$

No element is common.

Step 3: Conclusion

Not equal. Reason: $2 \in A$ but $2 \notin B$, and $3 \in A$ but $3 \notin B$.

(ii) $A = \{x : x \text{ is a letter in the word FOLLOW}\}$

$B = \{y : y \text{ is a letter in the word WOLF}\}$

Step 1: List elements

Word FOLLOW has letters: F, O, L, L, O, W

Set A (unique letters): $\{F, O, L, W\}$ …

Here are the common mistakes students make when solving Set Membership and Equality problems like this one, along with how to avoid each.

Mistake 1: Solving the quadratic equation incorrectly

The Error:

For part (i), students often solve $x^2 + 5x + 6 = 0$ and get the wrong roots — for example, writing $x = -2, -3$ but then forgetting the negative signs, or factoring as $(x+2)(x+3)$ but misreading the signs.

Why it happens:

Rushing through factorization or using the quadratic formula without double-checking signs.

How to Avoid:

• Always factor carefully: $x^2 + 5x + 6 = (x+2)(x+3) = 0$ gives $x = -2$ and $x = -3$.
• Write the solution set explicitly: $B = \{-2, -3\}$.
• Then compare with $A = \{2, 3\}$. Since $2 \neq -2$ and $3 \neq -3$, the sets are not equal.

Key takeaway: $A = \{2, 3\}$ and $B = \{-2, -3\}$ are different sets.

Mistake 2: Confusing “solution of equation” with “positive solution”

The Error:

Some students assume $x$ must be positive, so they discard $-2$ and $-3$ and write $B = \{\}$ or $B = \{2, 3\}$ incorrectly.

Why it happens:

A habit of only considering natural numbers or positive integers in earlier problems.

How to Avoid:

• Read the condition carefully: “$x$ is a solution of $x^2 + 5x + 6 = 0$” — no restriction to positive numbers.
• Always list all real solutions unless specified otherwise.

Mistake 3: Forgetting that sets are unordered and contain unique elements

The Error:

In part (ii), students list the letters of “FOLLOW” as $\{F, O, L, L, O, W\}$ and “WOLF” as $\{W, O, L, F\}$, then say they are not equal because the order or repetition differs.

Why it happens:

Confusing a list (ordered, with repeats) with a set (unordered, no repeats).

How to Avoid:

• For a set, remove duplicates and ignore order.
• $A = \{x : x \text{ is a letter in FOLLOW}\} = \{F, O, L, W\}$ (since L and O repeat).
• $B = \{y : y \text{ is a letter in WOLF}\} = \{W, O, L, F\}$.
• Both have exactly the same elements: $\{F, O, L, W\}$.
• Therefore, the sets are equal.

Key rule: Two sets are equal if and only if they contain the same elements, regardless of order or repetition.

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