Category Archives: College and Beyond

Introduction to Compound Propositions

Posted on by
Reply

In the previous post, we have learned about propositions. We learned that propositions are statements that are either true or false but not both. In this post, we are going to combine two or more propositions using words such as and, or, and if and then. Two or more propositions combined are called compound propositions and the words used to combined them are called logical connectors. We formalize our knowledge about compound propositions by the following definition.

Definition

A compound proposition is a proposition formed from simpler propositions using logical connectors or some combination of logical connectors. Some logical connectors involving propositions p and/or q may be expressed as follows: not p, p and q, p or q, if p then q.  Continue reading →

The Sum and Product of Roots Theorem

Posted on by
Reply

From the quadratic formula, we know that the numbers $latex r_1$ and $latex r_2$ are the roots of the quadratic equation $latex ax^2 + bx +c =0$ where $latex a \neq 0$ if and only if

$latex r_1 + r_2 = -\frac{b}{a}$

and

$latex r_1r_2 = \frac{c}{a}$.  Continue reading →

Triangle Inequality and Its Proof

Posted on by
1

Triangle Inequality states that for any real numbers a and b,

|a| + |b| ≥ |a + b|.

In proving this theorem, we use the definition of absolute value of a number. The absolute value of a number is x if x > 0 (*), –x  if x < 0 (**) and 0 if x = 0 (**). For instance, the absolute value of 5 is 5 since 5 > 0. On the other hand, the absolute value of -3 is -(-3) = 3. Clearly, the absolute value of 0 is 0. Triangle Inequality The geometric representation of triangle inequality is shown above (see Demystifying Triangle Inequality). It states that a triangle can only be formed if the combined length of any two sides is greater than that of the third sides. Using the definition of absolute value, we now prove the theorem. Continue reading →