## Thursday, September 18, 2008

### Geometry Chapter 2.5 - 2.8 Logic

Geometry chapter 2.5 Conditional Statements:

I) Vocabulary:
A. Conditional Statements - (if-then) - has two parts:

1. Hypothesis - (if)
2. Conclusion - (then)

Example: If it is noon in Georgia, then it is 9 am in California.

B) Converse Statement: Switch the hypothesis and conclusion of a Conditional Statement.

Example: If it is 9 am in California, then it is noon in Georgia.

C) Inverse Statement: negate both the hypothesis and conclusion of a Conditional Statement.

Example: If it is not noon in Georgia, then it is not 9 am in California.

1) Negation - write the negative of the statement.

Example: Statement: Angle A is acute.
Negation: Angle A is not acute or It is not true that angle A is acute.

D) Contrapositive Statement: Switch and negate both the Hypothesis and Conclusion of a Conditional Statement.

Example: If it is not 9 am in California, then it is not noon in Georgia.

E) Equivalent Statements: 2 statements that are both true or both false, they have the same truth value. Contrapositive statements is always equivalent to its Conditional statement. The converse statement is always equivalent to its inverse statement.

F) Counterexample: An example that shows that a conditional statement is false.

Example: If x2 = 25, then x = 5

A counterexample is x = (-5) because (-5)2 = 25 but 5 is not equal to (-5)

II) Postulates:

#5) Postulate 5: Through any 2 points there exists one line.
#6) Postulate 6: A line contains at least two points.
#7) Postulate 7: If two lines intersect, then their intersection is exactly one point.
#8) Postulate 8: Through any three noncollinear points there exists exactly one plane.
#9) Postulate 9: A plane contains at least three noncollinear points.
#10) Postulate 10: If two points lie in a plane, then the line containing them lies in the plane.
#11) Postulate 11: If two planes intersect, then their intersection is a line.

Geometry Chapter 2.6 Definitions and Biconditional Statements
I) Vocabulary:

A) Perpendicular: Two lines that intersect to form right angles.

B) Line Perpendicular to a Plane: A line that intersects a plane in a point and is perpendicular to every line that includes that point in the plane that intersects it.

C) Biconditional Statement: A statement that contains the words "if and only if" (iff) and is equivalent to writing a statement combining a conditional statement and its converse.

For the truth value of a biconditional statement to be true, both the conditional statemene and its converse have to have the same truth value.

Example: Conditional Statement: If two sides of a triangle are congruent, then the angles opposite them are congruent.
Converse Statement: If two angles of a triangle are congruent, then the sides opposite them are congruent.

Biconditional Statement: Two sides of a triangle are congruent if and only if the two angles of the triangle are congruent.

2.7 The Laws of logic -
patterns that are frequently used in drawing conclusions.

I. The law of Detachment -
A valid argument - uses a series of statements called premises that have known truth values to arrive at a conclusion.

If a conclusion is true (p → q ) and the hypothesis (p) is true, then the conclusion (q) is true.

Example: Given the following true statements, what can we conclude?
1. If adjacent angles are supplementary, then the angles form a linear pair.
2. Angle ABC and angle CBD are adjacent supplementary angles.

Conclusion: angle ABC and angle CBD form a linear pair.

II. The Law of Disjunctive Inference -
A. If a disjunction (p V q) is true and the disjunction (p) is false, then the other disjunction (q) has to be true.
B. If a disjunction (p V q) is true and the disjunction (q) is false, then the other disjunction (p) has to be true.

Example: Given the following true statements, what is a valid conclusion?
1. Paul is tall or Mort is short.
2. Paul is not tall.

Since Paul is not tall, so Mort has to be short.
Conclusion: Mort is short.

Example: Given the following true statements, what is a valid conclusion?
1. Carl listens to the radio or he cannot do his homework.
2. Carl cannot do his homework.

Since Carl cannot do his homework is true, then we have (p V true) is true. p could be either true or false so therefore:

Conclusion: No conclusion

Example: Given the following true statements, what sport does each person play?
1. Zach, Steve and David each play a different sport: basketball, soccer, or baseball. Zach made each of the following true statements:
2. I do not play basketball.
3. If Steve does not play soccer, then David plays baseball.
4. David does not play baseball.

What sport does each person play?
David does not play baseball. So therefore Steve does not play soccer is false for the 3rd statement to be true. This means that Steve does play soccer. This leaves David to play basketball and Zach to play baseball.

Example: Given the following true statements, which stock did Victoria sell yesterday?
1. Victoria owns stock in 3 companies: Alpha, Beta and Gamma.
2. Yesterday, Victoria sold her shares of Alpha or Gamma.
3. If she sold Alpha, then she bought more shares of Beta.
4. Victoria did not buy more shares of Beta.

Since Victoria did not buy more shares of Beta, then she did not sell Alpha. Since she did not sell Alpha, she had to sell her shares of Gamma.

Conclusion: She sold her shares of Gamma.