Assertion (A): Both cis `1,3`-dimethyl cyclobutane and trans `1-3`, dimethyl cyclobutane are optically inactivity
Reason (R ): cis-1,3- Dimethyl cyclobutane has the plane of symmetry, wheras trans form has the centre of symmetry.

Options

A. If both (A) and (R ) are true and (R ) is the correct explanation for (A).
B. If both (A) and (R ) are true and (R ) is not the true explanation for (A)
C. If (A) is true, but (R ) is false.
D. If both (A) and (R ) are false

Answer

To solve the question, we will analyze the assertion and the reason provided regarding the optical inactivity of both cis and trans isomers of 1,3-dimethylcyclobutane.

### Step-by-Step Solution:

1. **Understanding the Structures**:
- First, we need to draw the structures of both cis and trans 1,3-dimethylcyclobutane.
- **Cis-1,3-dimethylcyclobutane**: In this structure, both methyl groups (CH₃) are on the same side of the cyclobutane ring.
- **Trans-1,3-dimethylcyclobutane**: In this structure, the methyl groups are on opposite sides of the cyclobutane ring.

2. **Analyzing Optical Activity**:
- A compound is optically inactive if it has a plane of symmetry or a center of symmetry.
- **Cis-1,3-dimethylcyclobutane**: This structure has a plane of symmetry, meaning that if you were to cut the molecule in half along this plane, both halves would be mirror images of each other. Therefore, it is optically inactive.
- **Trans-1,3-dimethylcyclobutane**: This structure has both a plane of symmetry and a center of symmetry. The distance from one methyl group to the center of the molecule is equal to the distance from the center to the other methyl group. Hence, it is also optically inactive.

3. **Conclusion on Assertion and Reason**:
- The assertion states that both cis and trans 1,3-dimethylcyclobutane are optically inactive, which we have confirmed to be true.
- The reason states that cis-1,3-dimethylcyclobutane has a plane of symmetry, and trans-1,3-dimethylcyclobutane has a center of symmetry, which is also true.
- Since both the assertion and reason are true, and the reason correctly explains the assertion, we conclude that the correct option is that both the assertion and reason are true, and the reason is the correct explanation for the assertion.

### Final Answer:
Both assertion (A) and reason (R) are true, and R is the correct explanation for A.

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