Find the sum of (8a + 2b - 4)
2 answers:
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Statements can be proved by contrapositive, contradiction or by induction.
- <em>2.21 and 2.23 are proved by contrapositive</em>
- <em>2.22 is proved by induction</em>
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<u>2.21: If </u><u> is even, then n is even (By contrapositive)</u>
The contrapositive of the above statement is that:
<em>If n is odd, then </em><em> is odd</em>
Represent the value of n as:
, where
Take the cube of both sides
Expand
Group
Factor out 2
Assume w is an integer; where:
So, we have:
The constant term (i.e. 1) means that is odd.
Hence, the statement has been proved by contrapositive.
<em>i.e. If n is odd, then </em><em> is odd</em>
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<u>2.22 </u><u> is even, if and only if n is even</u>
We have: <u />
<u />
Assume that: for
So, we have:
Open bracket
Factorize
The factor of 2 means that is even.
<em>Hence, </em><em> is even, if and only if n is even </em>
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<u>2.22: </u><u> and </u>
<u>, then </u>
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To do this, we prove by contrapositive.
The contrapositive of the above statement is:
If and
, then
We have:
Substitute the values of s and t in:
Hence, by contrapositive:
If and
, then
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