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Draw the lattice diagram for the power set of with the set inclusion relation,
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Draw the lattice diagram for the power set of with the set inclusion relation,
Draw the diagram for the set of positive integers that are divisors of 30. Is this poset a Boolean algebra?
Draw a diagram of the lattice of subgroups of
Let be the set of positive integers that are divisors of 36. Define an order on by if Prove that is a Boolean algebra. Find a set such that is isomorphic to
Prove or disprove: is a poset under the relation if
False.
Draw the switching circuit for each of the following Boolean expressions.
(a)
(c)
Draw a circuit that will be closed exactly when only one of three switches and are closed.
Prove or disprove that the two circuits shown are equivalent.
Not equivalent.
Let be a finite set containing elements. Prove that Conclude that the order of any finite Boolean algebra must be for some
For each of the following circuits, write a Boolean expression. If the circuit can be replaced by one with fewer switches, give the Boolean expression and draw a diagram for the new circuit.
(a)
Prove or disprove: The set of all nonzero integers is a lattice, where is defined by
Let be a nonempty set with two binary operations and satisfying the commutative, associative, idempotent, and absorption laws. We can define a partial order on as in Theorem 19.14, by if Prove that the greatest lower bound of and is
Let be a group and be the set of subgroups of ordered by set-theoretic inclusion. If and are subgroups of show that the least upper bound of and is the subgroup generated by
Let be a ring and suppose that is the set of ideals of Show that is a poset ordered by set-theoretic inclusion, Define the meet of two ideals and in by and the join of and by Prove that the set of ideals of is a lattice under these operations.
Let be ideals in We need to show that is the smallest ideal in containing both and If and then is in For hence, is an ideal in
Let be a Boolean algebra. Prove each of the following identities.
and for all
If and then
for all
and
and (De Morgan's laws).
By drawing the appropriate diagrams, complete the proof of Theorem 19.30 to show that the switching functions form a Boolean algebra.
Let be a Boolean algebra. Define binary operations and on by
Prove that is a commutative ring under these operations satisfying for all
Let be a poset such that for every and in either or Then is said to be a .
Is a total order on
Prove that and are totally ordered sets under the usual ordering
(a) No.
Let and be posets. A map is if implies that Let and be lattices. A map is a if and Show that every lattice homomorphism is order-preserving, but that it is not the case that every order-preserving homomorphism is a lattice homomorphism.
Let be a Boolean algebra. Prove that if and only if for
A symmetric argument shows that
Let be a Boolean algebra. Prove that if and only if for all
Let and be lattices. Define an order relation on by if and Show that is a lattice under this partial order.