Q.Show that the set of letters needed to spell “ CA TARACT ” and the set of letters needed to spell “ TRACT” are equal

Concept: Set Equality via Set Difference

Two sets are equal if and only if they contain exactly the same elements (ignoring repetition and order).

First, identify the distinct letters in each word:

• "CATARACT" uses the letters $\{C, A, T, R\}$
• "TRACT" uses the letters $\{T, R, A, C\}$

Both sets contain the four letters $C$, $A$, $T$, and $R$. Even though "CATARACT" has repeated letters (three A's, two T's, two C's), a set records only distinct elements. Similarly, "TRACT" has repeated letters (T appears twice), but the set is $\{T, R, A, C\}$.

Since both sets have identical elements, they are equal: $\{C, A, T, R\} = \{T, R, A, C\}$.

The heart of this problem lies in understanding what a set is. A set is a collection of distinct objects — each element either belongs or it doesn't, and listing an element multiple times doesn't change the set. When we ask for "the set of letters needed to spell" a word, we're asking: which distinct letters appear at least once?

Let's identify the letters in each word carefully.

1. Find the distinct letters in "CATARACT".

   Write out the word: C-A-T-A-R-A-C-T.

   Now list each letter that appears, ignoring repetitions:

   • C appears (twice, but we count it once)
   • A appears (three times, but we count it once)
   • T appears (twice, but we count it once)
   • R appears (once)

   So the set of letters is $\{C, A, T, R\}$.

2. Find the distinct letters in "TRACT".

   Write out the word: T-R-A-C-T.

   List each distinct letter:

   • T appears (twice, but we count it once)
   • R appears (once)
   • A appears (once)
   • C appears (once)

   So the set of letters is $\{T, R, A, C\}$.

3. Compare the two sets.

   We have:

Sets have no inherent order, so $\{C, A, T, R\}$ and $\{T, R, A, C\}$ are the same set — they contain exactly the same four elements.

Method: Set Equality via Element Listing

We will use the Listing Method — write both sets explicitly by listing their distinct elements, then compare.

Step 1: Write the first set (from “CATARACT”)

Take the word CATARACT and list each distinct letter (ignore repetitions):

• C, A, T, A, R, A, C, T
• Distinct letters: C, A, T, R

So,

Step 2: Write the second set (from “TRACT”)

Take the word TRACT and list each distinct letter:

• T, R, A, C, T
• Distinct letters: T, R, A, C

So,

Step 3: Compare the two sets

• $S_1 = \{C, A, T, R\}$
• $S_2 = \{T, R, A, C\}$

Both contain exactly the same four letters. Order does not matter in a set, and repetitions are ignored.

Step 4: Conclusion

Since every element of $S_1$ is in $S_2$ and every element of $S_2$ is in $S_1$, we have:

Therefore, the set of letters needed to spell “CATARACT” equals the set of letters needed to spell “TRACT”.

Here are the most common mistakes students make on this Set Difference / Set Equality problem, along with how to avoid each.

Mistake 1: Counting repeated letters as separate elements

The error:

Students write the set for “CATARACT” as $\{C, A, T, A, R, A, C, T\}$ and then think the two sets are different because “CATARACT” has more letters.

Why it’s wrong:

A set does not contain duplicates. Each element appears only once. The letter $A$ appears three times in “CATARACT”, but in the set it is listed just once.

How to avoid:

Always write the roster form of a set by listing each distinct letter only once.

• For “CATARACT”: distinct letters are $C, A, T, R$ → set = $\{C, A, T, R\}$
• For “TRACT”: distinct letters are $T, R, A, C, T$ → set = $\{T, R, A, C\}$

Both sets are $\{A, C, R, T\}$ — equal.

Mistake 2: Confusing “set of letters” with “multiset” or “word”

The error:

Students think the order or frequency matters (e.g., “CATARACT” has 3 A’s, “TRACT” has 1 A, so they can’t be equal).

Why it’s wrong:

A set is unordered and has no multiplicity. The question explicitly says “set of letters”, not “list of letters” or “multiset”.

How to avoid:

Remember the definition:

A set is a collection of distinct objects, with no order.

If the problem meant frequency, it would say “multiset” or “bag of letters”.

Mistake 3: Forgetting to check both subset directions

The error:

Students only check that every letter in “TRACT” is in “CATARACT” (true), but don’t check the reverse.

Why it’s wrong:

Set equality requires $A \subseteq B$ and $B \subseteq A$. If you only check one direction, you might miss that “CATARACT” has a letter not in “TRACT” (it doesn’t, but the habit is important).

How to avoid:

Always do the two-way check:

1. Forward: Is every letter of “TRACT” in “CATARACT”?

   • $T$ ✓, $R$ ✓, $A$ ✓, $C$ ✓ → Yes.

2. Backward: Is every letter of “CATARACT” in “TRACT”?

   • $C$ ✓, $A$ ✓, $T$ ✓, $R$ ✓ → Yes.

Since both hold, the sets are equal.

Mistake 4: Writing the set in the wrong order and thinking order matters

The error:

A student writes $\{C, A, T, R\}$ for “CATARACT” and $\{T, R, A, C\}$ for “TRACT”, then says they are different because the order is different.

Why it’s wrong:

Sets are unordered. $\{C, A, T, R\} = \{T, R, A, C\}$.

How to avoid:

When comparing two sets, sort the elements alphabetically (or by any fixed rule) before comparing. This makes equality obvious.

• Sorted: both become $\{A, C, R, T\}$.

Mistake 5: Including spaces or punctuation as “letters”

The error:

The word “CATARACT” is written with a space in the problem: “CA TARACT”. Some students include the space as an element.

Why it’s wrong:

The problem says “letters needed to spell” — spaces are not letters.

How to avoid:

Ignore spaces, hyphens, apostrophes, etc. Only consider the alphabet characters (A–Z).

Quick Summary Table

Final takeaway:

The set of distinct letters in “CATARACT” is $\{A, C, R, T\}$.

The set of distinct letters in “TRACT” is also $\{A, C, R, T\}$.

They are equal because they contain exactly the same elements — no more, no less.