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No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$$$F(0)=7,\ F(1)=5,\ F(2)=3,\ F(3)=1$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$$$F(4)=6,\ F(5)=4,\ F(6)=2,\ F(7)=0$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.

No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.

No. Consider $$F(0)=7,\ F(1)=5,\ F(2)=3,\ F(3)=1$$ $$F(4)=6,\ F(5)=4,\ F(6)=2,\ F(7)=0$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.

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No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.

No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$

No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$ where a number represents its base-2 expansion, e.g. 6 represents $(1,1,0)$.

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No. Consider $$F(0)=0,\ F(1)=2,\ F(2)=1,\ F(3)=3$$ $$F(4)=0,\ F(5)=2,\ F(6)=1,\ F(7)=3$$