"Using Deductive Reasoning", 2.1 Conditionals and Converses

Well, yesterday, we did "Using Deductive Reasoning", or more specifically, 2.1 Conditionals and Converses.

Conditional: A statement in the form of an "If...then".

Example: If you brush your teeth, then they will be clean. "If you brush your teeth..." is the hypothesis (the "if" part), and "...then they will be clean." is the conclusion (the "then" part). The Hypothesis is denoted as a P, while the conclusion is denoted as a Q. With this, we can create what is called a truth table. The truth table allows us to reason every possibility. An example of the truth table for this example is down below.

T=Truth, F=False

P Q P>>Q

T F T

T F T

F T F

F F T

So basically, you just consider P and Q, and determine whether it is truth or false.

Example for F/T: If you (don't) brush your teeth, then they will be clean. This doesn't make sense, and is thus False.

Do note that the reasoning is only either a False or Truth.

Next, we learned about counter examples.

Counter Example: An example that disproves a statement. Only one is needed.

An example: If segment AB is congruent to segment BC, then B is the midpoint of segment AC.

Counter example:

A*--------------------------*B B*--------------------------*C

This diagram shows that even though AB is congruent to BC, that B is not always the midpoint of AC.

3rd, we learned what a converse statement is.

Converse: A switch between hypothesis and conclusion (P and Q). So instead of P>Q, Q>P.

Example: If x=5, then x^2=25. True. So if you flip it, if x^2=25, then x=5. This turns into a false.

Another: If ab=0, then a=0, or b=0. True. So if you flip it, if a=0, or b=0, then ab=0. True also. This leads to our next term that we learned.

Biconditional: A statement in the form of "If and only if". The abbreviation for it is Iff. This happens when the conditional and converse are true (P>Q and Q>P).

So basically, equality is represented here. <--> is the symbol for biconditional.

And finally, we learned the general forms of a conditional statement.

They are:

If...then.

P implies Q.

P only if Q.

P if Q.

And that sums it up for our lesson!

Before ending class, we got a project called "Geometry Around Us". It's due on Monday September 26. Basically, we need to make a poster with five photographs or physical examples of a math shape or solid.

Well that's all for September 6's class!

On a side note, the next scriber will be......

Sean Chung!

Until next time! Bye!