NIMCET 2007 — Mathematics PYQ

We can simplify the expression $A^{\prime }-B^{\prime }$ using standard laws and properties of sets.

Step 1: Understand the definition of Set Difference

For any two sets $X$ and $Y$, the set difference $X-Y$ is defined as the intersection of set $X$ with the complement of set $Y$:

$$X-Y=X\cap Y^{\prime }$$

Step 2: Apply the property to the given expression

Now, replace $X$ with $A^{\prime }$ and $Y$ with $B^{\prime }$ from image_21dcec.png into the definition:

$$A^{\prime }-B^{\prime }=A^{\prime }\cap (B^{\prime }{)}^{\prime }$$

Step 3: Simplify using the Double Complement Law

According to the double complement law in set theory, the complement of a complement set gives the original set back:

$$(B^{\prime }{)}^{\prime }=B$$

Substituting this back into our equation:

$$A^{\prime }-B^{\prime }=A^{\prime }\cap B$$

Step 4: Rearrange using the Commutative Law

The intersection of sets is commutative, meaning $A^{\prime }\cap B$ is the same as $B\cap A^{\prime }$:

$$A^{\prime }-B^{\prime }=B\cap A^{\prime }$$

Using the set difference property we started with ($B-A=B\cap A^{\prime }$), we can rewrite this as:

$$B\cap A^{\prime }=B-A$$

Therefore:

$$A^{\prime }-B^{\prime }=B-A$$

Conclusion

The simplified value of $A^{\prime }-B^{\prime }$ is equal to $B-A$.

Correct Option: (a) $B-A$