Q.From the sets given below, select equal sets : A = { 2, 4, 8, 12}, B = { 1, 2, 3, 4}, C = { 4, 8, 12, 14}, D = { 3, 1, 4, 2} E = {–1, 1}, F = { 0, a}, G = {1, –1}, H = { 0, 1}

Concept: Set Equality

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

Solution:

Compare each set by listing elements in a standard form:

• $A = \{2, 4, 8, 12\}$
• $B = \{1, 2, 3, 4\}$
• $C = \{4, 8, 12, 14\}$
• $D = \{3, 1, 4, 2\} = \{1, 2, 3, 4\}$
• $E = \{-1, 1\}$
• $F = \{0, a\}$
• $G = \{1, -1\} = \{-1, 1\}$
• $H = \{0, 1\}$ …

Understanding Set Equality

Set equality is one of the most fundamental concepts in set theory. Two sets are equal when every element of the first set is in the second, and every element of the second is in the first. 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 ${1, 2, 3}$.

The key is to look at membership: do the sets have identical rosters?

Step-by-Step Comparison

Let me organize the given sets first:

Now I'll systematically check for equality.

1. Compare sets with four elements: $A$, $B$, $C$, $D$

   Set $A = \{2, 4, 8, 12\}$ contains $8$ and $12$ but not $1$ or $3$.

   Set $B = \{1, 2, 3, 4\}$ contains $1$ and $3$ but not $8$ or $12$.

   Set $C = \{4, 8, 12, 14\}$ contains $14$ but not $2$.

   Set $D = \{3, 1, 4, 2\}$—reordering gives $\{1, 2, 3, 4\}$, which is exactly $B$.

   Therefore $B = D$.

2. Compare sets with two elements: $E$, $F$, $G$, $H$

   Set $E = \{-1, 1\}$ contains both negative and positive one. …

Method: Set Equality via Element Comparison

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

Steps:

1. List each set in ascending order (or any consistent order) to compare easily.
2. Check element-by-element — every element of one set must be in the other, and vice versa.
3. Mark pairs that are identical.

Applying the method:

Here are the common mistakes students make when identifying equal sets, along with clear strategies to avoid each.

Mistake 1: Ignoring the Order of Elements

The error:

Students see sets like $B = \{1, 2, 3, 4\}$ and $D = \{3, 1, 4, 2\}$ and think they are different because the numbers appear in a different sequence.

Why it’s wrong:

A set is defined only by its elements, not by the order they are listed.

$\{1, 2, 3, 4\}$ and $\{3, 1, 4, 2\}$ contain exactly the same four numbers — they are equal.

How to avoid:

Always sort or mentally rearrange the elements of each set into ascending order before comparing.

• $B = \{1, 2, 3, 4\}$
• $D = \{1, 2, 3, 4\}$ Now it’s obvious they match.

Mistake 2: Confusing “Equal” with “Same Number of Elements”

The error:

A student sees $A = \{2, 4, 8, 12\}$ and $C = \{4, 8, 12, 14\}$ — both have 4 elements — and assumes they are equal.

Why it’s wrong:

Equal sets must have exactly the same elements.

$A$ contains $2$ but not $14$; $C$ contains $14$ but not $2$. They are not equal.

How to avoid:

Check element-by-element after sorting. If even one element differs, the sets are not equal — regardless of size.

Mistake 3: Treating Repeated Elements as Different

The error:

A student writes $B = \{1, 2, 3, 4\}$ and sees another set listed as $\{1, 1, 2, 3, 4\}$ and thinks they are different because of the extra “1”.

Why it’s wrong:

In set notation, repetitions are ignored. $\{1, 1, 2, 3, 4\} = \{1, 2, 3, 4\}$.

(Though in this problem, no set has explicit repetitions — but the concept is tested often.)

How to avoid:

When comparing, remove duplicates mentally first. Only the distinct elements matter.

Mistake 4: Misreading Symbols or Variables

The error:

$F = \{0, a\}$ and $H = \{0, 1\}$ — a student sees “a” and thinks it means “1” or ignores it.

Why it’s wrong:

$a$ is a variable (or a letter), not the number $1$. Unless told $a = 1$, these sets are different.

Similarly, $E = \{-1, 1\}$ and $G = \{1, -1\}$ are equal (same elements, different order), but $F$ and $H$ are not equal.

How to avoid:

Treat letters as unknown symbols unless a specific value is given. Compare elements literally:

• $F$ has elements $0$ and $a$ …