Determine whether the statement is true, false, or sometimes true. Provide an example to justify your answer.
If both x and y are integers and x > y,
then x^2 > y^2.
not true
counterexample
5 > -6 ----> true
then 5^2 > (-6)^2 ??
25 > 36 ----> False
the statement is true only for positive values of x and y, so
it is "sometimes true"
To determine whether the statement is true, false, or sometimes true, we need to analyze the given statement and consider some examples.
The statement says that if both x and y are integers and x > y, then x^2 > y^2.
Let's consider an example to understand this statement.
Example:
Let x = 3 and y = 2.
In this case, x = 3 is greater than y = 2.
Now, let's calculate the squares of x and y:
x^2 = 3^2 = 9
y^2 = 2^2 = 4
From the example, we can observe that x^2 = 9 is indeed greater than y^2 = 4. Therefore, the statement "If both x and y are integers and x > y, then x^2 > y^2" is true in this example.
To generalize this, for any two integers x and y, where x > y, the statement will always hold true. This is because squaring both x and y will result in larger values for x^2 compared to y^2, given that x and y are positive integers.
Hence, we can conclude that the statement is true.