Let . Prove that .

We will begin by letting and letting . If , so , and and : since and , it follows that

This came from http://ashleymills.com/.

Let . Prove that .

We will begin by letting and letting . If , so , and and : since and , it follows that

This came from http://ashleymills.com/.

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Let . Prove that .

In order for because . If we assume that then and would both be and . . Also because then which creates a contradiction with the axiom above. Since , .

Let and . Prove that if and only if .

Let’s begin by letting and assume that . From this assumption it follows that . This means that . This creates a contradiction with the above statement and therefore . This proves the second half of the statement, and now we must prove the first half of the statement.

We will now let and assume that . From this assumption it follows that . This means that creating a contradiction with the above statement and therefore .

Let , . Prove that if and only if

Let and assume that . From this assumption it follows that , which means that , which can also be stated as which contradicts with the above assumption and therefore . Now if we let and assume that than it follows that . This means that which contradicts the above statement and therefore . This came from http://ashleymills.com/.