Question
upstudy study bank question image url

g) Use the truth table to show that the proportion is a tautology .
(4mks)
h) Prove that is divisible by 3 .
i) Given that find
i.
ii.

Ask by Mann Pearson. in Kenya
Dec 17,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

a) The proportion is a tautology.
b) is divisible by 3.
c) i. ; ii. .

Solution

a) To show that the proportion is a tautology using a truth table, we need to construct a truth table for the proposition and show that the proposition is always true regardless of the truth values of and .
Let’s construct the truth table for the proposition :
T T T T
T F F T
F T F T
F F T T
From the truth table, we can see that the proposition is always true, regardless of the truth values of and . Therefore, the proportion is a tautology.
b) To prove that is divisible by 3, we need to show that for any natural number , is divisible by 3.
Let’s prove this by mathematical induction:
Base case: For , , which is divisible by 3.
Inductive step: Assume that for some natural number , is divisible by 3. We need to show that is also divisible by 3.
Expanding :
Since is divisible by 3 by the inductive hypothesis, we have shown that is divisible by 3.
Therefore, by mathematical induction, is divisible by 3.
c) Given that , we need to find:
i.
ii.
i. To find , we substitute into the function :
ii. To find , we need to find the value of such that . This means solving the equation for .
Solving the equation:
Therefore, .

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Extra Insights

Consider the truth table for the expression . When is true, can either be true or false depending on . The expression states that if is true, then for the whole statement to remain true, must also be true. Thus, in all possible scenarios, the expression holds true, confirming it’s indeed a tautology.
To prove that is divisible by 3 for all natural numbers , consider modulo 3. For , ; for , ; and for , . Thus, in all cases, is divisible by 3.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy