# Making a Boolean algebra into a Partially Ordered Set

11b. A boolean algebra can be made into a partially ordered set by letting a≤b mean a=b.

Show that a≤b iff b= a + b

© BrainMass Inc. brainmass.com October 24, 2018, 6:10 pm ad1c9bdddfhttps://brainmass.com/math/boolean-algebra/making-a-boolean-algebra-into-a-partially-ordered-set-33361

#### Solution Summary

Making a Boolean algebra into partially ordered set is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.

Application of Set Theoretic Model of Sequences

1. Using the set-theoretic model of sequences, define the following operators,

giving their syntax and their semantics:

(a) overwrite: given any sequence, s, over a set X, any element, e, of X and any non-zero natural

number, n, return a sequence identical to s except that the element at position n is e.

For example, overwrite [a, b, c, d] f 3 = [a, b, f, d]

(b) insert: : given any sequence, s, over a set X, any element, e, of X and any non-zero natural

number, n, return a sequence identical to s except that e is at position n and every element of s

at position greater than n has been shifted right by one position.

For example, insert [a, b, c, d] f 3 = [a, b, f, c, d]

(c) take: given any sequence, s, over a set X and any non-zero natural number, n, return a

sequence consisting of the first n elements of s.

For example, take [a, b, c, d] 3 = [a, b, c] [5 marks]

(d) drop: given any sequence, s, over a set X and any non-zero natural number, n, return a

sequence containing all except the first n elements of s.

For example, drop [a, b, c, d] 3 = [d]

2. Using the theories of NAT and BOOL define the following operators as

conservative extensions:

(a) gte, the relation to which a pair of natural numbers belongs if the first is greater than or equalt

to the second.

For example, gte(5,5), gte(5,3) but not gte (3,5).

]

(b) sub, the partial operator that computes the difference between a pair of natural numbers (i.e.

subtracts the second from the first) and is defined only if the difference is a NAT.

For example, sub(5,3) = 2 but sub(3,5) is undefined [

(c) quotient, the partial operator that, given a pair of natural numbers, returns the highest natural

number by which the second can be multiplied without exceeding the first.

For example, quotient(7,3) = 1, quotient(3,7) = 0 but quotient(3,0) is undefined.

(d) remainder, the partial operator that, given a pair of natural numbers, returns the natural

number by which the first exceeds the product of the second and their quotient.

For example, remainder(7,3) = 4, remainder(3,7) = 3 but remainder(3,0) is undefined.