## Mathematical Proofs

###### This and the next few sections of the course provide various examples of proofs.  The proofs here are all direct proofs.  Initial assumptions are given and new results are derived used direct logical deduction.

Direct Proof Example 01

1/27/14

Running Time: 4:10

This video provides a simple example of a direct proof.  Assuming "a", "b" in R, and "a" less than "b" less than 0, we show that a^2 greater than b^2.  This proof is shown directly by using just the starting assumption, i.e. "a" less than "b" less than 0 and simple algebraic manipulations involving inequalities.

Direct Proof Example 02

1/28/14

Running Time: 9:16

In this problem we show that two sets are equivalent by showing that each set is a subset of the other.  The same problem is also worked in the subsequent video using set builder notation and logical operators.

Direct Proof Example 03

1/28/14

Running Time: 3:52

In this problem we show that two sets are equivalent by direct manipulation.  Definitions of set union, intersection, and the Distributive law are used to transform one set into the other.  The same problem was worked in the previous example, but a different approach was used.  In the previous example, the sets were shown to be equal be demonstrating that each set was a subset of the other.