Change the structure from 3SAT to 1in3 3SAT There is a variable set V = {x1,x2,x3} and clause set C1={x1,x2,-x3} C2={x1,-x1,-x2} C3={-x1,-x2,x3} C4={x2,x3,-x3}. For this structure, no matter each variable is positive or negative, the clause can always be TRUE by at least one literal is TRUE within each clause, just like 3SAT.
How can I change the variable gadgets and clause gadgets to make it no matter each variable is positive or negative, the clause can always be TRUE by only one literal is TRUE within each clause, just like 1 in3 3SAT?
A 1-in-3 SAT clause (A,B,C) can be expressed with the following 5 clauses in 3SAT. 

$(A ∨ B ∨ C) $

$(A ∨ \overline{B} ∨ \overline{C}) $

$(\overline{A} ∨ B ∨ \overline{C}) $

$(\overline{A} ∨ \overline{B} ∨ C) $

$(\overline{A} ∨ \overline{B} ∨ \overline{C}) $

Why? Consider the truth table over A,B,C. 

| A | B | C |   | A | B | C |
| 0 | 0 | 0 |   | --------- |
| 0 | 0 | 1 |   | 0 | 0 | 1 | 
| 0 | 1 | 0 |   | 0 | 1 | 0 | 
| 0 | 1 | 1 |   | --------- | 
| 1 | 0 | 0 |   | 1 | 0 | 0 |
| 1 | 0 | 1 |   | --------- | 
| 1 | 1 | 0 |   | --------- |
| 1 | 1 | 1 |   | --------- |


There are 8 lines. Each clause eliminates 1 line. We only want to allow (0,0,1), (0,1,0), (1,0,0) and eliminate the others.