Question
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.
(4mks)
h) Prove that
i) Given that
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.
.
b)
c) i.
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.
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,
.
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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.