Question 3. (10 points) A set X is said to be closed under an operation OP if, f

Question

Question 3. (10 points) A set X is said to be closed under an operation OP if, for any elements in X applying OP to them gives an element in X. For example, the set of integers is closed under multiplication because if we take any two integers, their product is also an integer For each of the sentences below, (1) first determine if it is a closure claim, then (2) determine if the sentence is true or false. For example, for the sentence "The product of any two prime numbers is not prime" is (1) not a closure claim but (2) is true. a. Concatenating two strings over the alphabet ? gives a string over the alphabet ?. b. The reversal of a string over the alphabet is a string over the alphabet . c. The intersection of two infinite sets of integers is an infinite set of integers. ble set of real numbers. e. The power set of a finite set of positive numbers is a finite set of positive numbers

Explanation / Answer

a. Yes, it is a closure claim.

And the closure claim is true because Concatenating two strings always gives a string.

b. Yes, it is also a closure claim.

And the closure claim is true because reversal of a string always gives a string.

c. Yes, it is also a closure claim.

But the closure claim is false because the intersection of two infinite set may be finite.

Consider the set of prime numbers and set of positive even numbers.

Intersection is {2} which is finite.

d. No, it is not a closure claim.

Also, the statement is false because the union of two countably infinite sets of real numbers may be countably infinite set.

Consider two sets of positive even numbers and positive odd numbers.

They are countable and their union is also countable.

e. Yes, it is a closure claim.

But it is false because power set is a set of sets instead of set of nummbers.

Please upvote. Thank you.

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