Download UNIT 2: Reasoning and Intro to Proofs

UNIT 2: Reasoning and Intro to Proofs
Objectives: (1) Recognize the hypothesis and the conclusion of an if-then statement
(2) State the converse, inverse, and contrapositive of an if-then statement.
(3) Use a counterexample to disprove an if-then statement.
(4) Understand the meaning of if and only if.
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A Conditional Statement is a logical statement that has two parts, a hypothesis (p) and a
conclusion (q). When a conditional statement is written in if-then form the part after the
word If is the hypothesis while the part after the then is called the conclusion. In a proof,
the hypothesis is our given information and the conclusion is the part we want to prove.
Example 1: If Jack has no homework then he is happy.
p: _____________________
q: _____________________
(Symbolically p represents hypothesis and q represents the conclusion)
Example 2: Each of the following conditionals are expressed w/o if-thens but can be put
into this form. Rewrite each statement in if-then form.
(A) Conditional: All dogs are canines.
If/then Form: If an animal is a dog then it’s a canine.
(B) Conditionals: 3 sided polygons are triangles.
If/then Form: If there is a 3 sided polygon then it’s a triangle.
(C) Conditional: The diagonals of a rectangle are congruent.
If/then Form: ________________________________________________
(D) Conditional: The sum of 2 odd integers is even.
If/then Form: _________________________________________________
An if-then conditional statement is false if an example can be found for which the
hypothesis is true but the conclusion is false. This example is called a
counterexample. You only need one counterexample to disprove a statement.
Example 3: Are the following conditionals true or false. If it is true write its given part
and the part you have to prove. If it is false find a counterexample.
(1) Two rectangles have equal areas if their perimeters are equal.
(2) Vertical angles are congruent.
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The negation (~) of a statement is the opposite of the original statement.
Statement 1: I am happy.
Statement 2: The dog is not hungry.
Negation 1: I am not happy.
Negation 2: The dog is hungry.
Example 4: Write the ~ of the following statement.
(1) The shape is a triangle.
(2) The angles are not obtuse.
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Related conditionals:
Conditional Statement:
If p then q
To write the converse of a conditional statement, switch the hypothesis and conclusion.
If q then p
To write the inverse of a conditional statement, negate both the hypothesis and conclusion
If ~ p then ~ q
To write the contrapositive of a conditional statement, negate both the hypothesis and the
conclusion.
If ~ q then ~ p
Example 5: Write the converse, inverse, and contrapositive of each conditional statement.
Tell whether each statement is true or false.
(A) If a polygon is equilateral then the polygon is regular.
Converse:
Inverse:
Contrapositive:
Equivalent Statements: A conditional statement and its contrapositive are either both true or
both false. Similarly, the converse and inverse of a conditional statement are either both true or
both false. Pairs of statements such as these are called equivalent statements.
Question: Can a statement and its converse be true? If the answer is yes can you think of an
example of such a case.
If a conditional and its converse are both true they can be combined into a single statement by
using the words “if and only if”. A statement that contains the words “if and only if” is called a
biconditional.
P if and only if q
Every definition can be written as a biconditional.
Definition: Perpendicular lines are two lines that intersect to form a right angle.
Biconditional: Two lines are perpendicular if and only if they intersect to form a right angle.
Homework:
State the hypothesis and the conclusion of each conditional.
(1) If 3x – 1 = 2 then x = 1.
(2) 4y = 20 implies y = 5.
(3)  1   2 if m  1 = m  2.
(4) Combine the conditional in exercise 3 into a single biconditional.
Provide a counterexample to show that each statement is false. You
may use words or draw a diagram.
(5) If AB  BC then B is the midpoint of AC .
(6) If a line lies in a vertical plane then the line is vertical.
(7) If a number is divisible by 4 then it is divisible by 6.
(8) If x 2 = 64 then x = 8.
(9) If ab < 0 then a < 0.
(10) If point G is on AB then G is on BA .
(11) If a four sided figure has four right angles, then it has four congruent sides.
(12) If a four-sided figure has congruent sides then it has four right angles.
Rewrite each pair of conditionals as a biconditional.
(13) If B is between A and C then AB + BC = AC.
If AB + BC = AC then B is between A and C.
Write the following biconditional as two conditionals that are converses of each other.
(14) Points are collinear if and only if they all lie in one line.
Tell whether each statement is true or false.
(15) If a polygon has five sides, then it is a regular pentagon(congruent sides and angles)
(16) If m  A is 85 degrees then the measure of the complement of  A is 5 degrees.
(17) Supplementary angles are always linear pairs.
(18) If a number is an integer then it is rational.
(19) If a number is a real number then it is irrational.
(20) Which statement has the same meaning as the given statement.
Given: You can go to the movies after you do your homework.
(A) If you do your homework then you can go to the movie afterwards.
(B) If you do not do your homework then you can go to the movies afterwards.
(C) If you cannot go to the movie afterwards then do your homework.
(D) If you are going to the movie afterwards then do not do your homework.
For the given statements, write the if-then form, the converse, the inverse, and the
contrapositive. Determine if each statement is true or false.
(21) The complementary angles add to 90 degrees.
(22) 3x + 10 = 16 because x = 2.
(23) Ants are insects.
(24) A midpoint bisects a segment.
Assume the following statement is true. Based on this fact determine whether the converse,
inverse, and contrapositive have to be true.
(25) All my students love math.