Blog 4

Todays lecture will be on truth tables. A truth table is a listing of all possible combinations of the individual statements of an argument as either true or false as well are the resulting truth values of the compound statement. In other words, by using a truth table we can determine the truth-values of an argument. The statement will be presented as either a ‘and’ , ‘or’ or an Ifàthen statement. For example: If you give me five dollars, then I will pay you back. There are several forms of truth tables for the different types of statements. These truth value forms include:

P	~P
T	F
F	T

~P                                             

  P and Q                                    (P^Q)

P	Q
T	F
T	T
F	T
F	F

P	Q	P^Q
T	F	F
T	T	T
F	T	F
F	F	F

                                   

P	Q	PvQ
T	T	T
T	F	T
F	T	T
F	F	F

P	Q	PàQ
T	T	T
T	F	F
F	T	T
F	F	T

  (PvQ) Disjunction                                 (PàQ) Conditional

            The last table is the conditional table. It is used for Ifàthen statements such as the example statement: If you give me five dollars, then I will pay you back. In this statement:

P=you give me five dollars

Q=I will pay you back

            The truth table for this statement would be set up at first like:

P	Q
T	T
T	F
F	T
F	F

The next step would to do the truth table of PàQ which would look like:

P	Q	PàQ
T	T	T
T	F	F
F	T	T
F	F	T

Thus the statements truth values are T, F, T, T. Thus the only instance in which the statement is false is when: If you give me five dollars, then I do not pay you back.

Fundamental Counting Principle

My name is professor Mary, and today I am going to teach you how to work with the Fundamental Counting Principle.

Objectives:

1.   Students will learn the definition of the Fundamental Counting Principle, and how to use it in basic math problems.

2. Students will get practice applying the Fundamental Counting Principle to real life situations.

Procedure:

1.    The definition of the Fundamental Counting Principle is: a way to figure out the total number of ways different events can occur.

2.    The Fundamental counting principle is meant to be a short cut when you want to find out how many ways something can be done, for example:

a.     How many different outfits can be made from 6 shirts and 5 pairs of pants?

                                               i.     To solve this you can simply multiply 5 pairs of pants x 6 shirts and get 30 out fit combinations.

3.   Let’s elaborate on why this works: to find the number of options someone has in any situation you simply multiple the number of variables by each other to find the total number of possibilities. This works because it is essentially the same as adding up all of the possible combinations.

4.  To see why the Fundamental Counting Principle is so important to math we will look at two ways to solve the following problem 

a.     How many sundae combinations can you make from having 3 kinds of ice cream: Vanilla, strawberry, and chocolate,  3 kinds of sauces: chocolate and caramel, and 3 kinds of toppings: chocolate chips, sprinkles, and fruit.

                                               i.     The first approach to solving this is to use the tree diagram, which is essentially just drawing out the options.

While the tree diagram is still useful, for problems with more variables it would be a lot harder to draw out and can get confusing and takes more time.

                      ii. The second way to solve this is to use the fundamental counting principle. By taking     the numbers you see one the side of the tree diagram in red: 3,2, and 3 we can multiply those to find the number of options rather than counting on the tree diagram.

   3 x 2 x 3 = 18

5.  Jenny wants to order a custom cake for her mom’s birthday. The bakery has 8 types of cake, 12 kinds of frosting, and 22 different decoration fonts. How many different cake combinations are possible for Jenny to choose from?

                     a.        8 x 12 x 22 = 2,112 different cake combinations.


This problem demonstrates why the Fundamental Counting Principle is so useful in saving math, and make it less likely for errors to occur when solving more complicated problems.