ECE 15A – Homework Solution # 1
1.
Expand the following into a polynomial having as few terms as possible:
a.
(x+y’x)(x+yz)
=(x+y’)(x+x)(x+yz)
from using law 3b
=x(x+y’)(x+yz)
where x+x=x from law 4b,
=x(x+yz)
where x(x+y’)=x from law 5a,
=x(x+yz)
again by using law 5a.
=x
b.
(x+y)(x’+y)(x+y’)(x’+y’)
=(y+xx’)(y’+xx’) by using law 3b
=yy’
where xx’=0 by law 6a
=0
where yy’=0 by law 6a
2.
Factor the following into linear factors:
a.
xy+zw
(x+zw)(y+zw) using law 3b
(x+z)(x+w)(y+z)(y+w) using law 3b
Factors x+z, x+w, y+z, and y+w are linear factors
b.
x+y(z+w)
(x+y)(x+z+w) using law 3b
Factors x+y, and x+z+w are linear factors
c.
abc+a’d
(a+a’d)(bc+a’d) using law 3b
(a+a’)(a+d)(b+a’d)(c+a’d) using law 3b
(a+d)(b+a’)(b+d)(c+a’)(c+d) using law 3b, where a+a’=1 by law 6b
Factors a+d, b+a’, b+d, c+a’ and c+d are linear factors
3.
Using Venn diagrams, show that if A and B are sets, then
1
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B
A
A+B
B
A
(A+B)'
B
A
A'
B
A
A'B'
A
B'
B
b.
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 Winter '08
 M
 Law, Addition, Elementary algebra, linear factors, Distributive law

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