Understanding If-Then Statements Part 1

In this post, we are going to examine the structure and truth table of the conditional statements or if-then statements. If-then statements are used often in mathematical proofs as well as real-life conversations. But before that, let us understand what proposition means.

A proposition is a declarative sentence that is either True (T) or false (F) but not both. If-then statements are composed of two propositions.

Let us consider the following statement by a father to his son.

“If you get an A in Calculus, then I’ll buy you a laptop.”

We can split this statement into two propositions p and q as follows.  Continue reading…

 

Another Proof of the Hypotenuse Leg Theorem

Given two triangles, if their hypotenuse are congruent, and one pair of their legs are congruent then the two triangles are congruent. In this post, we are going to prove this theorem.

In the figure below, ABC and DEF are right triangles with right angles at C and F, respectively.

hypotenuse leg 1

It is given that \overline{AB} \cong \overline{DE} and \overline{AC} \cong \overline{DF}. We are going to prove that \triangle ABC \cong \triangle DEFContinue reading…

 

Proof that for all real numbers a, |-a| = |a|

We can think of the absolute value of a number as its distance from 0. So, the absolute value of a, which is denoted by |a| is always greater than 0. In this post, we are going to prove that for all real numbers a, |-a| = |a|.

There are two possible cases: a \geq 0 and $latex b < 0$. (i) For $latex a \geq 0$, $latex since - a \leq 0$ $latex |-a| = -(-a) = a$ (since a is negative, we negate it to make it positive) $latex |a| = a$ (ii) For $latex a < 0$, since $latex - a > 0$,

|-a| = -a (since a is negative, we negate it to make it positive)
|a| = -a

By (i) and (ii), for any real number a,

|-a| = |a|.