Question

Suppose if we wanted to prove the following statement using the proof by contradiction, from what type of assumption should we start our proof with?
Statement: When x and y are odd integers, there does not exist an odd integer z such that \[x + y = z\].
(a) When x and y are odd integers, there does exist an odd integer z such that \[x + y = z\].
(b) When x and y are odd integers, there does exist an even integer z such that \[x + y = z\].
(c) When x and y are odd integers, there does not exist an odd integer z such that \[x + y = z\].
(d) None of these.

Answer 1

Hint: The given problem revolves around the concepts of proving the statements of geometry-type or algebraic-type solutions. As a result, to find/prove such kinds of statements by considering the/its contradiction particularly, hence solving the respective proof/solution by taking the opposite meaning of the given statement, to obtain the desired proof.

Complete answer:Since, we have given the statement that
“When x and y are odd integers, there does not exist an odd integer z such that \[x + y = z\].”
Where, odd integers means the numbers (or, integers) which are not divisible by \[2\] such as \[ - 7\], \[ - 5\], \[ - 3\], \[3\], \[5\], \[7\], & so on.
Since, as the question implies;
By assuming the contradiction of the statement which seems that the proof should be proved by assuming the given statement opposite to that of that is negative or positive to negative or positive assumption respectively (as per the statement implies).
For an instance,
Here, assuming that
“When x and y are odd integers, there does exist an odd integer z such that \[x + y = z\].”
Such that we have taken contradictory part of the given statement that the sum of the two odd integers equals to odd integers respectively.
Let us take one example by considering that two numbers as,
\[x = 5\] and, \[y = 7\] where both are odd
As a result, adding the terms, we get
\[x + y = 5 + 7\]
\[x + y = 12\]
Therefore, from the above solution/assumption, it seems that our statement (we have assumed) contradicts with the given statement.
Hence, it implies/states that we should start the proof by its negation of the respective given statement.
\[\therefore \]Option (a) is correct.

Note:
In these cases, remember the LOGICAL notation of the NEGATION statement such that if ‘p’ is any given statement then its negation (or, contradiction) can be represented as (i.e. mathematically) as \[ \sim p\] respectively. Also, remembering its property/rule that \[ \sim \left( { \sim p} \right) = p\] i.e. considering a negative statement as a positive one, so as to be sure of our final answer.