A prime number is a natural number greater than 1 which is not a product of two smaller natural numbers. Prove or disprove: For every integer q, if q > 7, then q can be written as q = (a + b * c) such that the following properties hold a and b are prime numbers, c is an integer greater than 1.

[Introduction to the Design and Analysis of Algorithms, 3rd Edition] Maxima search. A point (xi, yi) in the Cartesian plane is said to be dominated by point (xj , yj ) if xi ≤ xj and yi ≤ yj with at least one of the two inequalities being strict. Given a set of n points, one of them is said to be a maximum of the set if it is not dominated by any other point in the set. For example, in the figure below, all the maximum points of the set are circled. Design an efficient algorithm for finding all the maximum points of a given set of n points in the Cartesian plane. What is the time efficiency class of your algorithm?

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3. Two 7-digit integers (leading zeros allowed) are considered to be equivalent if one can be obtained from the other by rearranging its digits. For example, 0012234, 0312042, 3220014 and 2100243 are all considered equivalent, whereas 0012234 and 0001234 are not equivalent. How many nonequivalent 7-digit integers are there in which the digits 0, 2, 4 and 6 can each appear at most once?

Members of Set A = {Monday, Tuesday, Wednesday, Friday, Saturday, Sunday} What is the cardinality of St A ?

x Bird(x)=Can- Fly(x) This universal quantifier used here implies that Oa. All reptiles cannot fly Ob All that flies is a Bird All Birds can fly Oc. All chicken are birds Od.

Given the following recursively defined set S: Basis: 0 € S and 7 € S Recursive rule: if x ES and y = S, then: • x+yЄS x-yes Prove that every element in S is divisible by 7.

The correct statements are: If L₁ and L2 are not regular languages, then L₁ U L₂ cannot be regular. If L₁ and L₂ are regular languages, then L₁ L₂ = {w: w€ (L₁-L₂) or w€ (L₂-L₁)} is regular. It is possible that the union of a regular language and a nonregular language is regular. Every subset of a regular language is regular.

The correct statements are: A set is countable iff it is either finite or countably infinite. The set of natural numbers is countable. The power set of the set of natural numbers is countable. For any Σ Ø, Σ* is countably infinite.

PS: Please complete all of the subparts as they are all the same question and not in Handwritten paper.