In Exercises 35-42, use the laws in Definition 1 to show that the stated properties hold in every Boolean algebra.
39. Show that De Morgan’s laws hold in a Boolean algebra. That is, show that for allxandy,
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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- Please answer clear and none of them are infiniti or negative infiniti.arrow_forwardWhat is the induced fuzzy set of B1 in terms of the extension principle?B2?arrow_forwardConsider the following statement. For all sets A and B, (A ∪ Bc) − B = (A − B) ∪ Bc. An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.) Let any sets A and B be given. Then (A ∪ Bc) − B = (A ∪ Bc) ∩ Bc by the set difference law (1) = (Bc ∩ A) ∪ (Bc ∩ Bc) by the distributive law (2) = (Bc ∩ A) ∪ Bc by the idempotent law for ∪ (3) = (A − B) ∪ Bc by the set difference law (4) The absorption law is needed between steps (2) and (3).The double complement law is needed between steps (3) and (4).The complement law is needed between steps (2) and (3).The commutative law is needed between between steps (1) and (2).The commutative law is needed between between steps (3) and (4).arrow_forward
- For "x is in A intersect B" give an equivalent statement that uses "x is in A" and "x is in B". You may use Java's notation for the boolean operators, and "is in" for set membership.arrow_forwardFor all sets A and B, (A ∪ Bc) − B = (A − B) ∪ Bc. An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.) Let any sets A and B be given. Then (A ∪ Bc) − B = (A ∪ Bc) ∩ Bc by the set difference law (1) = (Bc ∩ A) ∪ (Bc ∩ Bc) by the distributive law (2) = (Bc ∩ A) ∪ Bc by the idempotent law for ∪ (3) = (A − B) ∪ Bc by the set difference law (4) The commutative law is needed between between steps (1) and (2). The complement law is needed between steps (2) and (3). The commutative law is needed between between steps (3) and (4). The absorption law is needed between steps (2) and (3). The double complement law is needed between steps (3) and (4).arrow_forwardwe have the following WFF of the quantificational calculus: ƎxƎy∀z (p(z) ∧ q(x,y) <-> (r(w) ∧ ∀w S(w,y)))How many occurrences does x,y,z and w have? and why How many bound and free occurrences does x,y,z and w have? and why I posted this question before but the answer is wrong. Thanksarrow_forward
- Give prove and also an example .arrow_forwardProve that that the natural numbers with the binary operation of multiplication (as defined in the video) forms a commutative monoid. Furthermore, prove that multiplication distributes over addition. Hint: First, you need to use induction to prove that given function f: X ⟶⟶X, (f^m)^n = (f^n)^marrow_forward
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