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3. If $x \in \{0, 1\}^n$ is a boolean string, let $\bar{x} \in \{0, 1\}^n$ denote the new string obtained by flipping the [10] bits in every coordinate in $x$. That is, $\bar{x}_i := 1 - x_i$ for every $i \in [n]$. For a concrete example, if $x = 101$ then $\bar{x} = 010$. Consider the following variation of the SAT problem. As input, you receive a boolean formula $F$ on variables $x_1, x_2, \dots, x_n$ (note that $F$ is not necessarily a CNF formula). The goal is to decide if $F$ has a satisfying assignment $x \in \{0, 1\}^n$ such that $\bar{x}$ is also a satisfying assignment. Prove that this variant of SAT is NP-Complete.

          3. If $x \in \{0, 1\}^n$ is a boolean string, let $\bar{x} \in \{0, 1\}^n$ denote the new string obtained by flipping the [10]
bits in every coordinate in $x$. That is, $\bar{x}_i := 1 - x_i$ for every $i \in [n]$. For a concrete example, if
$x = 101$ then $\bar{x} = 010$.
Consider the following variation of the SAT problem. As input, you receive a boolean formula $F$
on variables $x_1, x_2, \dots, x_n$ (note that $F$ is not necessarily a CNF formula). The goal is to decide
if $F$ has a satisfying assignment $x \in \{0, 1\}^n$ such that $\bar{x}$ is also a satisfying assignment. Prove
that this variant of SAT is NP-Complete.

$3. If x ∈{0, 1}^n is a boolean string, let x̅∈{0, 1}^n denote the new string obtained by flipping the [10] bits in every coordinate in x. That is, x̅i := 1 - xi for every i ∈ [n]. For a concrete example, if x = 101 then x̅ = 010. Consider the following variation of the SAT problem. As input, you receive a boolean formula F on variables x1, x2, …, xn (note that F is not necessarily a CNF formula). The goal is to decide if F has a satisfying assignment x ∈{0, 1}^n such that x̅ is also a satisfying assignment. Prove that this variant of SAT is NP-Complete.$

Added by Alex J.

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