7.1 problems

#2. This student paid me a visit, and I suggested that, while I didn't need these proven in all generality, I would like to see them proven for at least one case of, say, a < b < c (we actually need to consider permutations of these, too).

This student got a "tilda" because this should be more general than indicated. The number "0" is not our only choice for the "0" of the (prospective) Boolean algebra. Hence we might say "Let m be the zero; consider x=m-1:

x+m = (m-1) + m = max(m-1,m) = m

which is not equal to x: so m is not behaving like "0".

By the way: one may not invoke duality for these problems, unless duality applies to the min/max: we don't have a Boolean algebra, as part b shows! Hence, duality is not to be assumed.

#10.

The subtlety that this student picked up on that many others forgot is that one must justify the step

0'=1

or
1'=0

which was demonstrated in Practice 4. It is not an axiom itself....

Part c. shows a problem several students had: you need to explicitly indicate that we're talking about "circle-+" in the first and second lines - a binary operation different from the "+" of the Boolean algebra.


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