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3 votes

Let $\text{U}$ be a set and $\text{X, Y} \subseteq \text{U}$. Define operation twist by

$$

\operatorname{twist}\text{(X, Y)} =(\text{X} \cap \text{Y}) \cup(\overline{\text{X}} \cap \overline{\text{Y}}).

$$

Which of the following is/are True?

- $\overline{\operatorname{twist}\text{(X, Y)}}=\operatorname{twist}(\overline{\text{X}}, \text{Y})$
- $\overline{\operatorname{twist}\text{(X, Y)}}=\operatorname{twist}(\overline{\text{X}}, \overline{\text{Y}})$
- $\operatorname{twist}(\text{X, Y})=\operatorname{twist}(\overline{\text{X}}, \text{Y})$
- $\operatorname{twist}\text{(X, Y)}=\operatorname{twist}(\overline{\text{X}}, \overline{\text{Y}})$

2 votes

twist $(\mathrm{X}, \mathrm{Y})$ is nothing but $\text{X} \odot \text{Y}$ which is complement of $\text{X} \Delta \text{Y}$.

So, $\text{twist(X,Y)}$ is the complement of Symmetric difference of $\text{X,Y.}$

Note that symmetric difference is also called Exclusive Or.

We know that $\text{X} \odot \text{Y}=\overline{\text{X}} \odot \overline{\text{Y}}$, So, Option D is correct.

Similarly, we know that $\overline{\text{X} \odot \text{Y}}=\overline{\text{X}} \odot \text{Y}$, So, Option A is correct.

So, $\text{twist(X,Y)}$ is the complement of Symmetric difference of $\text{X,Y.}$

Note that symmetric difference is also called Exclusive Or.

We know that $\text{X} \odot \text{Y}=\overline{\text{X}} \odot \overline{\text{Y}}$, So, Option D is correct.

Similarly, we know that $\overline{\text{X} \odot \text{Y}}=\overline{\text{X}} \odot \text{Y}$, So, Option A is correct.