Logic and Counterexamples
Consider the following:
a: x is a prime number
b: x is a multiple of 3
If x = 24, then write the compound statement:
~a ∨ b
What is "24 is not a prime number or 24 is a multiple of 3"?
Write the biconditional of the following statement:
If an angle is a right angle, then it measures 90°.
What is "An angle is a right angle if and only if it measures 90°"?
Determine if the following argument is valid or invalid. If it is valid, through which law?
(1) If a polygon is a square, then it has four right angles.
(2) ABCD is a square.
"Therefore" ABCD has four right angles.
What is valid through the Law of Detachment?
Define p and q as follows:
p: a polygon is a hexagon
q: the sum of the interior angles is 720°
Write the biconditional p ↔ q in words.
What is "a polygon is a hexagon if and only if the sum of the interior angles is 720°"?
What is Ms. Swainston's favorite morning drink?
What is "Crio Bru"?
Choose the counterexample for the following statement:
If x is a whole number, then x3 > x2.
A. x = 12
B. x = 0
C. x = ½
D. x = 2
What is "x = 0"?
What is the relationship between the two statements below? (converse, inverse, contrapositive, or none of these)
(1) If it does not rain today, then I will wash the car.
(2) If I will wash the car, then it does not rain today.
What is converse?
Determine if the following argument is valid or invalid. If it is valid, through which law?
(1) If you understand logic, then you can reason correctly.
(2) If you can reason correctly, then you will pass your test.
"Therefore" If you do not pass your test, then you do not understand logic.
What is valid through the Law of Syllogism II?
Consider the following statements represented by p and q:
•p: Jalen is not a student at Mechanicsville High School
•q: Jalen will take the PSAT on October 16.
Write the contrapositive of the following statement in symbolic form:
If Jalen is a student at Mechanicsville HS, then he will take the PSAT on October 16th.
What is "~q -> p"?
Solve the following equation for x, showing all steps.
-2x + 5 = 13
p: Juan eats apples.
q: Juan drinks juice.
Write the following: p ∨ ∼q
A. Juan eats apples and Juan drinks juice.
B. Juan eats apples or Juan does not drink juice.
C. If Juan eats apples, then Juan does not drink juice.
D. Juan does not eat apples or drink juice.
What is B. Juan eats apples or Juan does not drink juice?
What is the relationship between the two statements below? (converse, inverse, contrapositive, or none of these)
(1)If I drive in the city, then I will need my GPS.
(2)If I do not need my GPS, then I do not drive in the city.
What is contrapositive?
Valid or Invalid: Using one of the laws, can you make a conclusion from the following statements? If valid, what is that conclusion?
(1) If a figure is a rectangle, then it has 4 right angles.
(2) If a figure is a square, then it is a rectangle.
Valid
What is "if a figure is a square, then it has 4 right angles?"
Or
What is "if a figure does not have 4 right angles, then it is not a square"?
Write the following statement as a conditional statement:
Every week has seven days.
What is "if it is a week, then it has seven days"?
Solve the following equation for a, showing all steps.
5 + 14a = 9a − 5
What is a = -2?
Given:
p: It will rain on Saturday.
q: We will play in the park.
Write the following statement using logic symbols.
If it does not rain on Saturday, then we will play in the park.
A. ∼q → p
B. p → q
C. ∼p ∧ q
D. ∼p → q
What is D. ∼p → q?
What is the relationship between the two statements below? (converse, inverse, contrapositive, or none of these)
(1) If I stop to smell the flowers, then I will sneeze.
(2) If I have allergies, then I will sneeze.
What is none of these?
Valid or Invalid: Using one of the laws, can you make a conclusion from the following statements? If valid, what is that conclusion?
If this wind keeps up, we will lose some trees.
We do not lose any trees.
What is "the wind did not keep up"?
Determine if the following statement can be written in "if, then" form. If so, what is the new statement?
All integers that are divisible by 10 are even.
What is "if it is an integer divisible by 10, then it is even"?
What is the cube root of 64?
What is 4?
Provide specific a counterexample for the following statement. If there is none, why?
Every number squared is positive.
None, any negative number squared turns positive as the negatives are multiplied together
What is the relationship between the two statements below? (converse, inverse, contrapositive, or none of these)
(1) If it is too cold, then I will not go fishing.
(2) If it is not too cold, then I will go fishing.
What is inverse?
Valid or Invalid: Using one of the laws, can you make a conclusion from the following statements? If valid, what is that conclusion?
(1) If a person is a librarian, then they read books.
(2) All friends of Dana's read books.
What is Invalid?
Using either the Law of Detachment or the Law of Contrapositive, make a valid conclusion from the symbolic statements. What law did you use?
~N -> M
~M
"Therefore"...
What is "N" using the Law of Contrapositive?
Solve the following equation for x, showing all steps.
7(2x − 1) − 11 = 6 + 6x
What is x = 3?