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Chapter 2
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Terms in this set (78)
inductive reasoning
A process that includes looking for patterns and making conjectures.
conjecture
An unproven statement that is based on observations.
counterexample
An example that proves a statement false.
conditional statement
A statement that can be written in if-then form.
hypothesis
The phrase following "if" in an if-then statement.
conclusion
The phrase following "then" in an if-then statement.
Venn diagram
A diagram that uses circles to display elements of different sets. Overlapping circles show common elements.
negation
If a statement is represented by p, then ~p is the _____ of the statement.
converse
Formed by switching the hypothesis and the conclusion in a conditional statement.
inverse
Formed by negating the hypotheses and conclusion.
contrapositive
Formed by switching and negating the hypothesis and conclusion.
deductive reasoning
USING FACTS, DEFINITIONS, ACCEPTED PROPERTIES AND THE LAWS OF LOGIC TO FORM A LOGICAL STATEMENT.
Detachment
Law of _____:
If p ➡ q is true, and p is true, then q must be true.
Syllogism
Law of _____:
If p ➡ q and q ➡ r are both true, then p ➡ r.
q ➡ p
Converse of p ➡ q.
~p ➡~q
Inverse of p ➡ q.
~q ➡~p
Contrapositive of p ➡ q.
If q, then p.
Converse of "If p, then q."
If not p, then not q.
Inverse of "If p, then q."
If not q, then not p.
Contrapositive of "If p, then q."
always
If a conditional is true, then its contrapositive is (always, sometimes, or never) true.
never
If a conditional is false, then its contrapositive is (always, sometimes, or never) true.
always
If the converse of a conditional is true, then the inverse of the conditional is (always, sometimes, or never) true.
never
If the converse of a conditional is false, then the inverse of the conditional is (always, sometimes, or never) true.
sometimes
If a conditional is true, then its converse is (always, sometimes, or never) true.
sometimes
If a conditional is true, then its inverse is (always, sometimes, or never) true.
converse
A biconditional is formed by the conjunction of a conditional and its ____________.
false
Conditional statements are (true or false) when the hypothesis is true, but the conclusion is false.
biconditional
Different ways you can write a __________:
p if and only if q
p iff q
b↔️q
~p ➡q
Inverse of p ➡~q
~q ➡p
Converse of p ➡~q
q ➡~p
Contrapositive of p ➡~q
Law of Detachment
Is the Law of Detachment or Law of Syllogism used to reach the conclusion?
Jared Jones will play football today if Vince Mercurio plays. Jones will not play football today. Conclusion: Thus, Mercurio will not play.
INDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? The Lions have lost their last four games. Thus, they will probably lose their next game.
DEDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? If you brush and floss your teeth daily then you will have fewer cavities. Marie brushes and flosses her teeth daily. Thus, she will have fewer cavities.
INDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? Four out of 5 times I beat Corey at pool and I'm going to play him tomorrow. So, I'll very likely win.
DEDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? An apple a day keeps the doctor away. Joe ate an apple every day. Dr. Dre stayed away.
DEDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? All Dogs are mammals. All mammals have kidneys. Therefore all dogs have kidneys.
DEDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? Euclid uses the division property of equality to solve 2x=140 and says "I have 70!"
INDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING? Euclid notices 2,19,36,53 and says "I have 70!"
logically equivalent
Statements with the same truth values are said to be ___.
you study for the test
What is the hypothesis of the conditional?
If you study for the test, then you will pass it.
you will pass it
What is the conclusion of the conditional?
If you study for the test, then you will pass it.
DEDUCTIVE
DEDUCTIVE OR INDUCTIVE REASONING?
If you are Justin Bieber, then you have a high pitched voice. If you have a high pitched voice, then you are a girl. Therefore, Justin Bieber is a girl.
FALSE - False Conditional (If |x| = 4, then x = -4. Counterexample is x = 4.)
IS THE BICONDITIONAL TRUE OR FALSE?
|x| = 4 iff x = -4.
FALSE - True hypothesis and false conclusion (Counterexample is x = 2.)
IS THE CONDITIONAL TRUE OR FALSE?
If 2x + 3 = 4x -1, then x = 3.
TRUE - False hypothesis and false conclusion
IS THE CONDITIONAL TRUE OR FALSE?
If apples are dogs, then cats are birds.
TRUE - True hypothesis and true conclusion.
IS THE CONDITIONAL TRUE OR FALSE?
If x + 3 = 4, then x = 1.
TRUE (True Conditional and True Converse)
IS THE BICONDITIONAL TRUE OR FALSE?
Segments are congruent if and only if they have the same length.
FALSE - False Converse (If an angle is acute, then its measure is 64 degrees. Counterexample is 70 degrees.).
IS THE BICONDITIONAL TRUE OR FALSE?
The measure of an angle is 64 degrees iff the angle is acute.
9 and 16 (25 is odd) or
25 and 16 (41 is odd) or...
The sum of two square numbers is an even number. Find a counterexample proving the conditional is false.
If I sleep, then I snore.
I snore when I sleep. Write as a conditional statement.
TRUE
Conditional statements are (true or false) when the hypothesis is false, but the conclusion is false.
TRUE
Conditional statements are (true or false) when the hypothesis is true, but the conclusion is true.
TRUE
Conditional statements are (true or false) when the hypothesis is false, but the conclusion is true.
False. Possible counterexample: -3 x -5 = 15
Is the conditional true or false?
The product of two negative numbers is negative.
you go to American Pie
What is the hypothesis of the conditional?
You order macaroni and cheese pizza if you go to American Pie.
Taylor orders boring cheese pizza
What is the conclusion of the conditional?
Taylor orders boring cheese pizza if she goes to American Pie.
deductive
When you use the Laws of Detachment and Syllogism, you are using ___ reasoning.
if and only if
What does "iff" represent?
Law of Syllogism
Is the Law of Detachment or Law of Syllogism used to reach the conclusion?
If Ickies glomp, then Schmoodles plop. If Schmoodles plop, then Fadoodoes twerk. Conclusion: If Ickies glomp, then Fadoodoes twerk.
supplementary
Two angles are______ if and only if the sum of their measures is 180 degrees.
If an animal is a lab, then it is a dog.
Write a conditional statement from the Venn diagram.
(What was he thinking????????)
Conclusion: Emili was not asked to Homecoming.
If Emili was asked to Homecoming, then she was asked in a really cute way. Emili was asked by text. Conclusion?
contrapositive
A conditional and its (converse, inverse, or contrapositive) always have the same truth value.
inverse
The converse and (inverse or contrapositive) of a conditional always have the same truth value.
If you study, then you have more free time. :)
Use the Law of Syllogism to reach a conclusion:
If you do well on a test, then you will not retest.
If you study, then you will do well on a test.
If you do not retest, then you have more free time.
Alex's car might not start.
Draw a conclusion from the given information:
If you leave your car lights on overnight, then your car battery will drain. If your battery is drained, then your car might not start. Alex left his car lights on last night.
x = 3 or x = -3
Solve: |x| = 3.
x = 5 or x = -5
Solve: x² = 25.
No
Is the conjecture valid by the Law of Syllogism?
Given: If a figure is a square, then the figure is a rectangle. If a figure is a square, then it is a parallelogram. Conjecture: If a figure is a square, then it is a parallelogram.
Yes
Is the conjecture valid by the Law of Syllogism?
Given: If an element is an alkali metal, then it reacts with water. If an element is in the first column of the periodic table, then it is an alkali metal. Conjecture: If an element is in the first column of the periodic table, then it reacts with water.
No
Is the conjecture valid by the Law of Detachment?
If you want to go on a field trip, you must have a signed permission slip. Zola has a signed permission slip. Conjecture: Zola wants to go on a field trip.
Yes
Is the conjecture valid by the Law of Detachment?
If it is colder than 50 degrees, then Tom wears a sweater. It is 46 degrees today. Conjecture: Tom wears a sweater today.
If lines are parallel, then they do not intersect.
Write a conditional statement from the sentence : "Parallel lines do not intersect."
If I am not hungry, then I do not eat.
Given: If I am hungry, then I eat.
What is the inverse of the conditional?
If I do not eat, then I am not hungry.
Given: If I am hungry, then I eat.
What is the contrapositive of the conditional?
If I eat, then I am hungry.
Given: If I am hungry, then I eat.
What is the converse of the conditional?
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