## Transcribed Text

5. In this problem we shall prove the inductive step in the proof by math-
ematical induction of the formula in (1). Let P(n) be the statement
that (1) is true for any finite sets A1- A2,..., An. Suppose P(4) true.
Let A1 1.Az.An,As.As be finite sets. Then
(Un/-((u)/und
DAING
Explain why each of the above equalities of sets is true. Prove that the
set on the last line is equal to
04JCNs
Hint: The nonempty subsets of N° are exactly of two types:
(b) J =10(5} where CNA In this case, we have two subtypes:
i. J= IU {5}, where 0# ICNA,
ii. J={5}.
6. Let f Y be an injection. Let IO E X. Define g X - (Io} -
y {f(=o)} by
g(x) /(x) for zex-(20).
2
(a) Prove that is well defined. That is, for any x € X (Io)
g(x) {f(=o)}
(b) Prove that g is an injection.
7. Let P(n) be the statement: Ifm E2+ and / N.. N.. is an injection,
then m n. Suppose E that P(k) is true. Let me z+
and
let / N, + NA+1 be an injection. Define
by g(i) f(i) for if Nm-1:
(a) Prove that a is an injection
(b) Using the Addition Principle, prove that |Nx+1 {((m)}| k:
(c) Using g and part (b), show that there exists an injection h :
(d) Prove that P(k 1) is true.
8.
Let f x Y be surjection. Let ID E X. Define h x {=o} + Y
by
h(x) f(x) for X (Io}
Suppose that h is not surjection. Define g X-{x > {((xx)}
by
9(2)=h(7)=f(f) x {xo}
(a) Prove that g is well defined. That is, for any x E x - (Io)
(b) Prove that g is surjection.
9. Let P(n) be the statement: If m EZ+ -> N.. is asurjection
then m Sn. Suppose AEZ+ is such that P(k) is true. Let me z+
and let f: Ne+1 Nm be surjection. Define
by h(i) /(i) for Ng
(a) Prove that if is surjection, then 772 3
(b) Suppose h is not a surjection. Prove that g Nr N, -
{/(k+1)} by
g(i)=h(i) =f(i)forieNa
is a well defined surjection.
(c) Assuming the same hypothesis as (b), prove that there exists a
surjection j
(d) Prove that P(A 1) is true.

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