Question:
Which of the following is not same as \((A\Delta B)\cap C\)?
Which of the following is not same as \((A\Delta B)\cap C\)?
Show Hint
The identity
\[
A\Delta B=(A\cup B)\cap(A\cap B)^c
\]
is extremely useful in simplifying set theory questions.
Updated On: Jun 11, 2026
- \((A\cap C)\Delta(B\cap C)\)
- \((A\cap B\cap C)^c\cap((A\cup B)\cap C)\)
- \(((A\cap B^c)\cap C)\cup((B\cap A^c)\cap C)\)
- \((A\cap B)^c\cap C\)
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The Correct Option is D
Solution and Explanation
Concept:
Symmetric difference is defined as
\[
A\Delta B
=
(A-B)\cup(B-A)
\]
\[
=
(A\cap B^c)\cup(B\cap A^c).
\]
Step 1: Expand \((A\Delta B)\cap C\).
\[ (A\Delta B)\cap C = [(A\cap B^c)\cup(B\cap A^c)]\cap C \] \[ = ((A\cap B^c)\cap C) \cup ((B\cap A^c)\cap C). \] Hence option (C) is identical.
Step 2: Check option (A).
\[ (A\cap C)\Delta(B\cap C) \] also expands to the same expression. Hence (A) is equivalent.
Step 3: Check option (B).
Using \[ A\Delta B=(A\cup B)\cap(A\cap B)^c, \] option (B) is equivalent.
Step 4: Check option (D).
\[ (A\cap B)^c\cap C \] does not require membership in either \(A\) or \(B\). Hence it is larger than \[ (A\Delta B)\cap C. \] Therefore it is not equivalent. \[ \boxed{(D)} \]
Step 1: Expand \((A\Delta B)\cap C\).
\[ (A\Delta B)\cap C = [(A\cap B^c)\cup(B\cap A^c)]\cap C \] \[ = ((A\cap B^c)\cap C) \cup ((B\cap A^c)\cap C). \] Hence option (C) is identical.
Step 2: Check option (A).
\[ (A\cap C)\Delta(B\cap C) \] also expands to the same expression. Hence (A) is equivalent.
Step 3: Check option (B).
Using \[ A\Delta B=(A\cup B)\cap(A\cap B)^c, \] option (B) is equivalent.
Step 4: Check option (D).
\[ (A\cap B)^c\cap C \] does not require membership in either \(A\) or \(B\). Hence it is larger than \[ (A\Delta B)\cap C. \] Therefore it is not equivalent. \[ \boxed{(D)} \]
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