1. The equality operation on naturals is correctly computed by the following pseudo-Java method: boolean equals (natural...

70.2K

Verified Solution

Question

Programming

1. The equality operation on naturals is correctly computed bythe following pseudo-Java method:
boolean equals (natural x, natural y) {
if (zero(x)) return zero(y);
return equals(pred(x), pred(y));
}
2. Suppose we define a function f from naturals to naturals asfollows. We define f(0) to be 0, and for any natural x we definef(Sx) to be 1. Then this function satisfies the equation f(x Â· y) =f(x) Â· f(y), for any naturals x and y.
3.
Suppose I want to prove \"for all x: S0 + x = Sx\". The followingis valid:
Ordinary induction on all naturals x.
Base case: x = 0, the desired formula becomes \"S0 + 0 = S0\".This is true by the base case of the definition of addition.
Inductive hypothesis: S0 + x = Sx
Inductive goal: S0 + Sx = S(Sx)
Proof of inductive step: By the inductive case of the definitionof addition, S(S0 + x) = S0 + Sx. By the inductive hypothesis,applied inside the parentheses, the left-hand side of this equationis equal to S(Sx)). So the inductive goal is true.
Please help with True or False Questions

Answer & Explanation Solved by verified expert

3.6 Ratings (537 Votes)

The solutions for the above questions is given below and if you feel any problem then feel free to ask 1 True because the function is of type boolean and returning bool also the function is taking two arguments as natural x and y And in the first See Answer

Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:

Unlimited Question Access with detailed Answers
Zin AI - 3 Million Words
10 Dall-E 3 Images
20 Plot Generations
Conversation with Dialogue Memory
No Ads, Ever!
Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!

Become a Member