Slide 38
Slide 38 text
P =
“the
product
of
x and
y is
even”
Q =
“at
least
one
of
x or
y must
be
even”
To
prove,
P implies
Q,
we
prove
the
contrapositive:
NOT(Q) implies NOT(P)
To
prove
an
implication,
assume
left
side,
show
right:
Assume NOT(at
least
one
of x and
y must
be
even)
By
the
meaning
of
NOT:
both
x and
y are
not
even
By
the
Odd-‐Even
Lemma:
both
x and
y are
odd
By
the
definition
of
odd:
there
exists
integers
k,
m,
such
that
x = 2k + 1 and
y = 2m + 1