14 let f r r r r defined by f x y x y x y and let g f

14. Let f : R ⇥ R ! R ⇥ R defined by f ( x, y ) := ( x + y, x - y ) , and let g = f ◦ f . Prove that g is a bijection.
15. Let X be a finite nonempty subset of N . Let | X | = n , and assume that n ≥ 2 . Let W = 2
X
- { ; } . Define
a function q : W ! X such that for any A 2 W , q ( A ) is the maximum natural number in A .
(a) Prove that q is surjective.
(b) Prove that q injective = ) 2
n
- 1  n . (c) Prove by ordinary induction on n that for all n ≥ 2 , n + 1 < 2
n
.
(d) Prove that q is not injective.
9

16. Let X and A be sets such that ; 6 = A ✓ X , and let f : X ! X be a function ( Warning: the domain and
the codomain are the same set). Recall the definition of the direct image of A through f :
f ( A ) = { y 2 X | 9 a 2 A, y = f ( a ) }
Now denote f ( A ) by B . B is another subset of X , so now we can talk about the direct image f ( B ) ; in fact
we can write f
(
f ( A )
)
instead of f ( B ) . Prove that f ( A ) ✓ A = ) f
(
f ( A )
)
✓ f ( A ) .
17. Consider sequences of n bits, n ≥ 1 , in which there are exactly k 0’s, where k < n , and every 0 is
immediately followed by a 1. How many such sequences are there? (Give the answer and an explanation
of how you figured it out. No proofs required.)