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. d. The union of two countably infinite sets of real numbers is an uncountable set of real numbers. e. The power set of a finite set of positive numbers is a finite set of positive numbers.

Explanation / Answer

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. In addition the statement, which is closure, will be TRUE however the statement may be TRUE if not closure.

This is closure as the string, which is produced, is another string containing the alphabets, which are there in other two strings. The statement is also TRUE.

This is closure as the string produced as a result of reversal is also a string in the same alphabets. The statement is also TRUE.

This is closure and TRUE, as intersection of two infinite sets of integers will produce infinite set of integers. However all infinite sets intersection may not be infinite always.

The statement is TRUE but closure as resulting set produced may not be having elements in the same set.

The statement is TRUE as power of any +ve number will be +ve number and the same number of elements in the resulting set. This is not closure the power of a number may not be in the same set.

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