Question 1

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If $\frac { 1 } { \mathrm { n } }$ is not an integer, then $\mathrm { n }$ is not an integer.
A) If $\mathrm { n }$ is an integer, then $\frac { 1 } { \mathrm { n } }$ is an integer. false
B) If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. false
C) If $\frac { 1 } { n }$ is an integer, then $n$ is an integer. true
D) If $\mathrm { n }$ is an integer, then $\frac { 1 } { n }$ is an integer. true

Accepted Answer

The answer of Write the contrapositive of the statement. Then...

Question 2

Write the indicated statement. Use De Morgan's Laws if necessary.
-If it is a cat, then it catches birds.
Inverse
A) If it doesnʹt catch birds, itʹs not a cat.
B) If itʹs not a cat, it doesnʹt catch birds.
C) Not all cats catch birds.
D) If it catches birds, itʹs a cat.

Accepted Answer

The inverse of a conditional statement "If P, then Q" is "If not P, then not Q." The original statement is "If it is a cat, then it catches birds." The inverse is "If it is not a cat, then it does not catch birds."

Question 3

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If the sum of the interior angles of a polygon does not measure 180°, then the polygon is not a Triangle.
A) If the polygon is a triangle, then the sum of the interior angles of the polygon measures 180°. true
B) If the polygon is not a triangle, then the sum of the interior angles of the polygon do not measure 180°. true
C) If the polygon is not a triangle, then the sum of the interior angles of the polygon do not measure 180°. false
D) If the polygon is a triangle, then the sum of the interior angles of the polygon measures 180°. false

Accepted Answer

The contrapositive of the given statement is correctly represented in choice A. The contrapositive of a conditional statement "If P, then Q" is "If not Q, then not P." Therefore, the contrapositive of the given statement is "If the polygon is a triangle, then the sum of the interior angles of the polygon measures 180°," which is true because by definition, the sum of the interior angles of a triangle is always 180°.

Question 4

Write an equivalent sentence for the statement.
-If the box is in the mail, then it should be here by tomorrow. (Hint: Use the fact that p → q is Equivalent to ~p ∨ q.)
A) If the box is in the mail, then it should not be here by tomorrow.
B) The box is not in the mail or it should be here by tomorrow.
C) The box is in the mail so it should be here by tomorrow.
D) The box is not in the mail and it should not be here by tomorrow.

Accepted Answer

The logical statement "If the box is in the mail, then it should be here by tomorrow" can be represented as $p \rightarrow q$. The equivalent form using the hint provided is $\neg p \vee q$, which translates to "The box is not in the mail or it should be here by tomorrow."

Question 5

Write an equivalent sentence for the statement.
-If it is raining, you take your hat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.)
A) It is not raining and you take your hat.
B) It is not raining and you do not take your hat.
C) If it is raining, you do not take your hat.
D) It is not raining or you take your hat.

Accepted Answer

The logical statement "If it is raining, you take your hat" can be represented as $p \rightarrow q$, which is logically equivalent to $\neg p \vee q$. This translates to "It is not raining or you take your hat," matching choice D.

Question 6

Write the indicated statement. Use De Morgan's Laws if necessary.
-If I pass, Iʹll party.
Contrapositive
A) If I party, then I passed.
B) If I donʹt party, I didnʹt pass.
C) If I donʹt pass, I wonʹt party.
D) Iʹll party if I pass.

Accepted Answer

The contrapositive of "If I pass, I'll party" is "If I don't party, then I didn't pass," which negates and reverses the original statement.

Question 7

Write an equivalent sentence for the statement.
-An number n is divisible by 3 if and only if the sum of the digits of n is divisible by 3. (Hint: Use The fact that (p → q) ∧ (q → p) is equivalent to p ↔ q.)
A) The sum of the digits of n is divisible by 3 or the number n is divisible by 3.
B) If a number n is divisible by 3 then the sum of the digits of n is divisible by 3, and if the sum of the digits of n is divisible by 3 then the number n is divisible by 3.
C) If a number n is divisible by 3 then the sum of the digits of n is divisible by 3, or if the sum of the digits of n is divisible by 3 then the number n is divisible by 3.
D) A number n is divisible by 3 and the sum of the digits of n is divisible by 3.

Accepted Answer

Option B correctly rephrases the given statement using the logical equivalence of "if and only if" (↔) as a combination of two conditional statements (→ and ←), which matches the hint provided.

Question 8

Write the indicated statement. Use De Morgan's Laws if necessary.
-If the moon is out, then we will start a campfire and we will roast marshmallows.
Inverse
A) If we start a campfire and we roast marshmallows, then the moon is out.
B) If the moon is not out, then we will not start a campfire or we will not roast marshmallows.
C) If we do not start a campfire or we do not roast marshmallows, then the moon is not out.
D) If the moon is not out, then we will start a campfire but we will not roast marshmallows.

Accepted Answer

The inverse of a conditional statement "If P, then Q" is "If not P, then not Q." Here, P is "the moon is out," and Q is "we will start a campfire and we will roast marshmallows." The inverse is "If the moon is not out, then we will not start a campfire or we will not roast marshmallows," which uses De Morgan's Laws to negate the conjunction.

Question 9

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If a line does not intersect a circle in more than one point; then the line is tangent to the circle.
A) If a line is tangent to a circle, then the line does not intersect the circle in more than one point. true
B) If a line is tangent to a circle, then the line intersects the circle in more than one point. false
C) If a line is not tangent to a circle, then the line intersects the circle in more than one point. true
D) If a line is not tangent to a circle, then the line intersects the circle in more than one point. false

Accepted Answer

The answer of Write the contrapositive of the statement. Then...

Question 10

Determine which, if any, of the three statements are equivalent.
-I. If tomorrow is not Monday, then today is not Sunday.
II. Today is not Sunday and tomorrow is not Monday.
III. It is not the case that today is Sunday and tomorrow is Monday.
A) I and II are equivalent
B) II and III are equivalent
C) I and III are equivalent
D) None are equivalent

Accepted Answer

The answer of Determine which, if any, of the three...

Question 11

Determine which, if any, of the three statements are equivalent.
-I. If the house is not white, then the house is not small.
II. The house is not white and the house is small.
III. It is false that the house is white or that the house is not small.
A) II and III are equivalent
B) I, II and III are equivalent
C) I and III are equivalent
D) None are equivalent

Accepted Answer

The answer of Determine which, if any, of the three...

Question 12

Write the indicated statement. Use De Morgan's Laws if necessary.
-If it is love, then it is blind.
Contrapositive
A) If it is blind then it is love.
B) If it is not blind, then it is not love.
C) If it is blind then it is not love.
D) It is blind if it is love.

Accepted Answer

The contrapositive of "If it is love, then it is blind" is "If it is not blind, then it is not love." This is because the contrapositive inverts and negates both the hypothesis and the conclusion of the original statement.

Question 13

Write the indicated statement. Use De Morgan's Laws if necessary.
-If $x = 7$, then $x ^ { 2 } = 49$
Converse
A) If $x ^ { 2 } \neq 49$, then $x \neq 7$.
B) If $x ^ { 2 } = 49$, then $x = 7$
C) If $x \neq 7$, then $x ^ { 2 } \neq 49$.
D) If $x = 7$, then $x ^ { 2 } = 49$.

Accepted Answer

The converse of a conditional statement switches the hypothesis and conclusion. The original statement is "If $x = 7$, then $x ^ { 2 } = 49$", so its converse is "If $x ^ { 2 } = 49$, then $x = 7$".

Question 14

Write the indicated statement. Use De Morgan's Laws if necessary.
-If you like me, then I like you.
Converse
A) I like you if you donʹt like me.
B) If you donʹt like me, I donʹt like you.
C) I donʹt like you if you donʹt like me.
D) If I like you, then you like me.

Accepted Answer

The converse of a conditional statement "If P, then Q" is "If Q, then P." Therefore, the converse of "If you like me, then I like you" is "If I like you, then you like me."

Question 15

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If a is not an even counting number, then the product of a and the counting number b is not an Even counting number.
A) If the product of a and the counting number b is an even counting number, then a is an even counting number. false
B) If the product of a and the counting number b is an even counting number, then a is an even counting number. true
C) If the product of a and the counting number b is an even counting number, then a is not an even counting number. true
D) If the product of a and the counting number b is an even counting number, then a is not an even counting number. false

Accepted Answer

The contrapositive of the given statement is correctly represented in option A. The contrapositive of "If a is not an even counting number, then the product of a and the counting number b is not an even counting number" is "If the product of a and the counting number b is an even counting number, then a is an even counting number." However, the statement is false because if b is even, the product can be even regardless of whether a is even or odd.

Question 16

Write the indicated statement. Use De Morgan's Laws if necessary.
-If the chores are done, then we will go to the carnival and we will eat cotton candy.
Contrapositive
A) If we do not go to the carnival and we do not eat cotton candy, then the chores are not done.
B) If the chores are not done, then we will not go to the carnival or we will not eat cotton candy.
C) If we do not go to the carnival or we do not eat cotton candy, then the chores are not done.
D) If we go to the carnival and we eat cotton candy, then the chores are done.

Accepted Answer

The contrapositive of a conditional statement "If P, then Q" is "If not Q, then not P." The original statement is "If the chores are done (P), then we will go to the carnival and we will eat cotton candy (Q)." The contrapositive is "If we do not go to the carnival or we do not eat cotton candy (not Q), then the chores are not done (not P)." This matches choice C.

Question 17

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If 10 does not divide the counting number, then 5 does not divide the counting number.
A) If 5 divides the counting number, then 10 divides the counting number. true
B) If 10 divides the counting number, then 5 divides the counting number. true
C) If 10 divides the counting number, then 5 divides the counting number. false
D) If 5 divides the counting number, then 10 divides the counting number. false

Accepted Answer

The contrapositive of a statement "If P, then Q" is "If not Q, then not P". The original statement is "If 10 does not divide the counting number, then 5 does not divide the counting number." The contrapositive is "If 5 divides the counting number, then 10 divides the counting number." This statement is false because a number can be divisible by 5 without being divisible by 10 (e.g., 5, 15, 25, etc.).

Question 18

Write an equivalent sentence for the statement.
-If you are mistaken then you are wrong, and if you are wrong then you are mistaken. (Hint: Use The fact that (p → q) ∧ (q → p) is equivalent to p ↔ q.)
A) You are mistaken or you are wrong.
B) You are mistaken and wrong.
C) You are mistaken if and only if you are wrong.
D) You are mistaken if you are wrong.

Accepted Answer

The given statement establishes a bidirectional condition between being mistaken and being wrong, implying that one occurs if and only if the other does. This is directly represented by the logical equivalence of $p \leftrightarrow q$, making option C the correct rephrasing.

Question 19

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false.
-If the triangle is not equilateral, then the three sides of the triangle are not equal.
A) If three sides of a triangle are not equal, then the triangle is not equilateral. true
B) If three sides of a triangle are equal, then the triangle is equilateral. false
C) If three sides of a triangle are not equal, then the triangle is not equilateral. false
D) If three sides of a triangle are equal, then the triangle is equilateral. true

Accepted Answer

The contrapositive of the statement "If the triangle is not equilateral, then the three sides of the triangle are not equal" is "If the three sides of a triangle are equal, then the triangle is equilateral." This statement is true.

Question 20

Determine which, if any, of the three statements are equivalent.
-I. Jan is well or Jan is still recovering.
II. If Jan is still recovering, then Jan is not well.
III. If Jan is well, then Jan is not still recovering.
A) I and II are equivalent
B) II and III are equivalent
C) I, II, and III are equivalent
D) None are equivalent

Accepted Answer

Statements II and III are logically equivalent because they both establish a mutual exclusivity between Jan being well and Jan still recovering. If Jan is still recovering (II), then she is not well, and if Jan is well (III), then she is not still recovering. Both imply that Jan cannot be both well and still recovering at the same time.

Quiz 3: Logic

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If $\frac { 1 } { \mathrm { n } }$ is not an integer, then $\mathrm { n }$ is not an integer.

Write the indicated statement. Use De Morgan's Laws if necessary. -If it is a cat, then it catches birds. Inverse

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If the sum of the interior angles of a polygon does not measure 180°, then the polygon is not a Triangle.

Write an equivalent sentence for the statement. -If the box is in the mail, then it should be here by tomorrow. (Hint: Use the fact that p → q is Equivalent to ~p ∨ q.)

Write an equivalent sentence for the statement. -If it is raining, you take your hat. (Hint: Use the fact that p → q is equivalent to ~p ∨ q.)

Write the indicated statement. Use De Morgan's Laws if necessary. -If I pass, Iʹll party. Contrapositive

Write an equivalent sentence for the statement. -An number n is divisible by 3 if and only if the sum of the digits of n is divisible by 3. (Hint: Use The fact that (p → q) ∧ (q → p) is equivalent to p ↔ q.)

Write the indicated statement. Use De Morgan's Laws if necessary. -If the moon is out, then we will start a campfire and we will roast marshmallows. Inverse

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If a line does not intersect a circle in more than one point; then the line is tangent to the circle.

Determine which, if any, of the three statements are equivalent. -I. If tomorrow is not Monday, then today is not Sunday. II. Today is not Sunday and tomorrow is not Monday. III. It is not the case that today is Sunday and tomorrow is Monday.

Determine which, if any, of the three statements are equivalent. -I. If the house is not white, then the house is not small. II. The house is not white and the house is small. III. It is false that the house is white or that the house is not small.

Write the indicated statement. Use De Morgan's Laws if necessary. -If it is love, then it is blind. Contrapositive

Write the indicated statement. Use De Morgan's Laws if necessary. -If $x = 7$ , then $x ^ { 2 } = 49$ Converse

Write the indicated statement. Use De Morgan's Laws if necessary. -If you like me, then I like you. Converse

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If a is not an even counting number, then the product of a and the counting number b is not an Even counting number.

Write the indicated statement. Use De Morgan's Laws if necessary. -If the chores are done, then we will go to the carnival and we will eat cotton candy. Contrapositive

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If 10 does not divide the counting number, then 5 does not divide the counting number.

Write an equivalent sentence for the statement. -If you are mistaken then you are wrong, and if you are wrong then you are mistaken. (Hint: Use The fact that (p → q) ∧ (q → p) is equivalent to p ↔ q.)

Write the contrapositive of the statement. Then use the contrapositive to determine whether to conditional statement is true or false. -If the triangle is not equilateral, then the three sides of the triangle are not equal.

Determine which, if any, of the three statements are equivalent. -I. Jan is well or Jan is still recovering. II. If Jan is still recovering, then Jan is not well. III. If Jan is well, then Jan is not still recovering.