Page 292 - ABCTE Study Guide_Neat

Example: for all real numbers x

We may be tempted to say that this statement is “always” true, because by choosing different values of x,
like -2 and 3, we see that:

This statement will be true for all negative values of x, and most positive values of x. However, this

2
statement isfalse for . To see this, let x = ; then we get x = . In this case, it is not true

that , which means that our statement is “sometimes” true—it is true for some real numbers x.

Example: For all primes , x is odd.

This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not
odd, it would be divisible by two, which would make x not prime.

Review

• Know and be able to identify common mathematical errors, such as circular arguments,
assuming the truth of the converse, assuming the truth of the inverse, making faulty
generalizations, and faulty use of analogical reasoning.
• Be familiar with direct proofs and indirect proofs (proof by contradiction).
• Be able to work with problems to identify “always,” “sometimes,” and “never”
statements.