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.

Chapter 2 Logic

Chapter 2 - Logic

Who are Leibniz, Boole, DeMorgan?

2 - 1 Sentences, Statements, and Truth Value

I. Logic - is the science of reasoning.
- help us to determine if a statement is true, false, or uncertain
- (the truth value of the statement)

The statements that we will use will be mathematic sentences

3 + 5 = 8 This is a true mathematical sentence

A midpoint of a line segment divides the line segment into 2 congruent parts.
This is a true mathematical sentence.

A pentagon is a six-sided polygon. This is a false mathematical sentence.

4 + 6 = 10. This is a false mathematical sentence.

II. Nonmathematical Sentences and Phrases
Sentences that do not state a fact, such as questions, commands, or exclamations
are not sentences that we use in the study of logic.

Example:

4 - 3 This is not a mathematical sentence

Go to your room! This is not a mathematical sentence.

III. Open Sentences - sentences that contain a variable.

Example:

x + 2 = 16 open sentence; variable is x

He ran the football in for a touchdown. open sentence; variable is he

17 - x = 9 open sentence; variable is x

Domain - is the input or the replacement set - the set of elements that are possible replacements for the variable.

Example:
x + 2 = 16 with the domain {5, 6, 7, 8, 9 }

Solution Set: the element or elements from the domain that make the open sentence true.

Therefore, the solution set is {8} because when x = 8, then 17 - 8 = 9 is true.

IV. Statement or closed sentence - can be judged to be true or false (no variables) - the truth value is either true (T) or false (F)

V. Negations of a statement always has the opposite truth value of the given statement.

Example: A frog is a snake. (This is a false statement).
Negation: A frog is not a snake. (This is a true statement).

Example: A triangle is not a polygon with 4 sides. (this is a true statement)
Negation: A triangle is a polygon with 4 sides. (this is a false statement)

to write this negation, you may have wanted to say:
A triangle is not not a polygon with 4 sides. As this doesn't make grammarical sense, we change the double negative to a positive.

VI. You can let a statement be represented by a lowercase letter.

Usually we use p, q, r, or s

If "p" is true, then not "p" or "~p" is false.

~p represents symbolically: not p.

Example:

Let p: Summer follows spring.

~p: Summer does not follow spring.

p is true so ~p is false

2 - 2 Conjunctions: is a compound statement formed by combining two simple statements using the word "and". From the table above, you see the symbol for and looks like an upside down V.

Example:

Let p: A dog is an animal.

Let q: A trumpet is a brass instrument.

What is the sentence for p and q?

A dog is an animal and a trumpet is a brass instrument.

What is the truth value for the following (we will negate the different parts of the sentence)

1. A dog is an animal and a trumpet is a brass instrument. (T and T = T)

2. A dog is an animal and a trumpet is not a brass instrument. (T and F = F)

3. A dog is not an animal and a trumpet is a brass instrument. (F and T = F)

4. A dog is not an animal and a trumpet is not a brass instrument. (F and F = F)

2 - 3 Disjunctions - is a compound statement formed by combining two simple statements using the word "OR" and the symbol for or is V.

Example:

Let p: January is the first month of the year.

Let q: Breakfast is a meal.

What is the sentence for p or q?

January is the first month of the year or breakfast is a meal.

What is the truth value for the following (we will again negate each part of the sentence)

January is the first month of the year or breakfast is a meal. (T or T = T)

January is the first month of the year or breakfast is not a meal. (T or F = T)

January is not the first month of the year or breakfast is a meal. (F or T = T)

January is not the first month of the year or breakfast is not a meal. (F or F = F)

Example:

Buffalo Bills is a football team.

Buffalo Sabres is a hockey team.

the or statement would be:

Buffalo Bills is a football team or Buffalo Sabres is a hockey team.

What is the truth value of this sentence?

True

Example:

Let our Set A = {1, 2, 3}

Set B = {2, 4, 6}

What is:

Set A V Set B = {1, 2, 3, 4, 6}

you include all the elements in both sets for "or"

Set A and Set B = {2}

you include only the elements that are in both set A and set B. This is what you called the intersection of the two sets.

I. Complement - is that which is not included in the set

Example:

the complement of Set A = {4, 6}

the complement of Set B = {1, 3}

II. Inclusive "or" - when we use the word "or" to mean that one or both of the simple sentences are true. This is the truth table we have used in this class.

2 - 4 Conditional - is a compound statement formed by using the words:

if ... then

symbolically: if p then q is p → q

"p" is the hypothesis or premise or antecedent

"q" is the conclusion or consequent

Example:

If today is Tuesday, then tomorrow is Wednesday.

Hypothesis: today is Tuesday

Conclusion: tomorrow is Wednesday

Other ways to write conditionals.

1. If today is Tuesday, then tomorrow is Wednesday.

2. Today is Tuesday implies that tomorrow is Wednesday.

3. When today is Tuesday, tomorrow is Wednesday.

Geometry - Chapter 1 - Essentials of Geometry

Chapter 1 - Essentials of Geometry

1-1 Undefined Terms:
A. Undefined terms: their meaning is accepted without definition
1. Point: definition - is that which has no part, a dot.
2. Set: is a collection of objects
3. Line - is breathless length;
4. Straight line: is a line that lies evenly with the points on itself.
5. Plane: is a set of points that form a flat surface extending indefinitely in all directions.

1-2 The real numbers and their Properties
B. Every real number corresponds to a point on a number line and every point on the number line corresponds to a real number.

1-3 Definitions, lines, and line segments

A. Definition: is a statement of the meaning of the term

B. Collinear set of points: is a set of points all of which lie on the same straight line.

C. Noncollinear set of points: is a set of three or more points that DO NOT all lie on the same straight line.

D. Distance between two points - every point on a line corresponds to a real number called its coordinate. To find the distance between any two points, find the absolute value of the difference between the coordinates of the two points.

If point A is at -2 and point D is at 1, what is the distance between A and D?

take the absolute value of (-2 - 1) = the absolute value of (-3) = 3

E. Order of points on a line
1. Betweenness: B is between A and C if and only if A, B, and C are distinct collinear points and AB + BC = AC

also, this is the partition postulate: the part + the part = the whole

2. Line Segment: or segment, is a set of points consisting of two endpoints on a line, called endpoints, and all of the points on the line between the endpoints.

1-5 Rays and Angles:

Definition: two points, A and B, are on one side of a point P if A, B, and P are collinear and P is not between A and B.

Half-line: every point on a line divides the line into 2 opposite set of points called half-lines.

Definition: A ray consists of a point (endpoint) on a line and all points on one side of the point.

Definition: Opposite rays: are two rays of the same line with a common endpoint and no other point in common.

Point B is the vertex, BA is a side of the angle and BC is the other side of the angle.

Definitions:

The measure of an angle is the number of degrees in the angle.

A straight angle is an angle that is the union of opposite rays.

Its degree measure is 180 degrees

Acute Angles - is an angle whose degree measure is greater than 0 degrees and less than 90 degres.

Right angles - is an angle whose degree measure is 90 degrees.

Obtuse angle - is an angle whose degree measure is greater than 90 degrees and less than 180 degrees.

Note:If you bisect a straight angle, you get two 90 degree angles or 2 right anglesIf you bisect an obtuse angle, you get two acute angles.

1-6 More angle definitions

Congruent angles are angles that have the same measure.

Definition: A bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides that angle into two congruent angles.

Given angle ABC with angle bisector BD, we can conclude that angle ABD is congruent to angle CBD.

If angle ABC measures 70 degrees, what is the measure of angle ABD? 35 degrees

Example: Given angle QRS with angle bisector RT, if the measure of angle QRS = 10x, and the measure of angle SRT = 3x + 30, what is the measure of angle QRS?

we know: measure of angle QRT = measure of angle SRT + measure of angle QRS

and since the angle QRS is bisected by ray RT, then angle QRT = angle SRT

so 3x + 30 = measure of angle SRT,

therefore

3x + 30 + 3x + 30 = 10x

6x + 60 = 10x

60 = 4x

15 = x

measure of angle QRS = 10x = 10 (15) = 150 degrees

1-7 Triangles:

I) Vocabulary:

A) Base angles - the two angles adjacent to the base of an isosceles triangle.

B) Vertex angles - the angle opposite the base of an isosceles triangle.

II) Theorems:

A) Base Angles Theorem: if two sides of a triangle are congruent, then the angles opposite them are congruent.

B) Converse of the Base Angles Theorem: if two angles of a triangle are congruent, the the sides opposite them are congruent.

Corollaries:

A) If a triangle is equilateral, then it is equiangular.

B) If a triangle is equiangular, then it is equilateral.

I) Vocabulary:

A) Triangle - is a figure formed by three segments joining three noncollinear points.

B) Classification of triangles:

1) By sides:

a) Equilateral Triangle - has 3 congruent sides

b) Isosceles Triangle - has at least 2 congruent sides

c) Scalene Triangle - has no congruent sides

2) By angles:

a) Acute Triangle - has 3 acute angles

b) Equiangular Triangle - has 3 congruent angles that measure 60 degrees each

c) Right Triangle - has one right angle and 2 acute angles

d) Obtuse Triangle - has one obtuse angle and 2 acute angles

C) Vertex - each of the three points joining the sides of a triangle (plural - vertices)

D) Adjacent Sides - in a triangle, two sides sharing a common vertex.

E) Right triangles have 2 sides that form the right angle called the legs. The side opposite the right angle is the hypotenuse of the triangle.

F) Interior angles - when the sides of a triangle are extended, other angles are formed. the three original angles are the interior angles.

G) Exterior angles - the angles that are adjacent to the interior angles.

H) Theorem:

1) Triangle sum theorem - the sum of the measures of the interior angles of a triangle is 180 degrees.

2) Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

I) Corollary to a theorem - is a statement that can be proved easily using the theorem.

Example: The acute angles of a right triangle are complementary.

Example: given Triangle ABC with side BC extended through point D, if angle A = 65 degrees and angle ACD = 2x + 10 and angle B = x, solve for x.
angle A + angle B = angle ACD65 + x = 2x + 1055 = x