Question

If $ A = \{a, b, c, d\} $ and $ B = \{d, c, b, a\} $ then

• A. $ A \neq B $
• B. $ A = B $
• C. $ A \cap B = \phi $
• D. $ A \cup B = \phi $

Hint:

In sets, order does not matter. $\{a,b,c\} = \{c,b,a\}$ always.

Answer 1

The Correct Option is B

Concept: In set theory, two sets are said to be equal if they contain exactly the same elements, regardless of the order in which the elements are written. Mathematically, \[ A = B \iff (\forall x)(x \in A \Leftrightarrow x \in B) \] Also, sets are unordered collections, so rearranging elements does not change the set.

Step 1: Write the elements of set $A$.
\[ A = \{a, b, c, d\} \] So the elements are: $a, b, c, d$.

Step 2: Write the elements of set $B$.
\[ B = \{d, c, b, a\} \] So the elements are: $d, c, b, a$.

Step 3: Compare both sets.
Check each element:
• $a \in A$ and $a \in B$
• $b \in A$ and $b \in B$
• $c \in A$ and $c \in B$
• $d \in A$ and $d \in B$ Thus, both sets contain exactly the same elements.

Step 4: Evaluate each option.

• (1) $A \neq B$ → Incorrect, since all elements are identical.
• (2) $A = B$ → Correct, same elements in both sets.
• (3) $A \cap B = \phi$ → Incorrect, because intersection is: \[ A \cap B = \{a, b, c, d\} \neq \phi \]
• (4) $A \cup B = \phi$ → Incorrect, since union contains all elements: \[ A \cup B = \{a, b, c, d\} \] Conclusion: Since both sets contain identical elements, \[ \boxed{A = B} \]