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Define the taxi norm on |R^n and prove that it is a norm.

x = (x[1],x[2],...,x[n])

||x|| = +(k=1 to k=n) |x[k]|

||x|| = +(k=1 to k=n) |x[k]|

What does it mean that two norms are equivalent?

Let W be a vector space. Let f,g be two norms on W.

\/A>0 \/B>0 /\x:-W A*f(x) <= g(x) <= B*f(x)

It is an equivalence relation.

\/A>0 \/B>0 /\x:-W A*f(x) <= g(x) <= B*f(x)

It is an equivalence relation.

Let a,b be real numbers.

sqrt(a*a + b*b) <= ???

sqrt(a*a + b*b) <= ???

sqrt(a*a + b*b) <= |a| + |b|

Let D be a connected subset of |C.

Let f,g:D->|C be continuous.

Suppose that /\z:-D exp(f(z)) = exp(g(x)).

What can we say about functions f and g?

Let f,g:D->|C be continuous.

Suppose that /\z:-D exp(f(z)) = exp(g(x)).

What can we say about functions f and g?

\/k:-|Z /\z:-D f(z) = g(z) + k*2*PI*i

page 37 in OLDTIMER

page 37 in OLDTIMER

A,M,B are arbitrary sets.

Express differently in three ways:

(M\A)\B = ??? = ??? = ???

Express differently in three ways:

(M\A)\B = ??? = ??? = ???

(M\A)\B = M\(AuB) = (M\A)n(M\B) = (M\B)\A

A,M,B are arbitrary sets.

Express it differently in three ways:

M\(AuB) = ??? = ??? = ???

Express it differently in three ways:

M\(AuB) = ??? = ??? = ???

M\(AuB) = (M\A)n(M\B) = (M\A)\B = (M\B)\A

A,M,B are arbitrary sets.

Express it differently in three ways:

(M\A)n(M\B) = ??? = ??? = ???

Express it differently in three ways:

(M\A)n(M\B) = ??? = ??? = ???

(M\A)n(M\B) = (M\A)\B = M\(AuB) = (M\B)\A

Let A,B subsets of X. If GcX, then 1(G) denotes the
characteristic function of G.

Express differently:

1(AuB) = ???

Express differently:

1(AuB) = ???

1(AuB) = max {1(A),1(B)}

Let A,B be subsets of X. If GcX, then 1(G) denotes the
characteristic function of G.

Express differently:

max {1(A),1(B)} = ???

Express differently:

max {1(A),1(B)} = ???

max {1(A),1(B)} = 1(AuB)

Let A,B subsets of X. If GcX, then 1(G) denotes the
characteristic function of G.

Express differently:

A c B <=> ???

Express differently:

A c B <=> ???

A c B <=> 1(A)<=1(B)