CISP 440 Final Exam questions and
answers already passed
A universal existential statement is a statement - correct answer ✔✔that is universal because its
first part says that a certain property is true for all objects of a given type, and it is existential
because its second part asserts the existence of something.
a. Is ( 5, −5 ) = ( −5, 5 ) ?
b. Is ( −2−4−2−4 , (−2)3 ) = ( 3636 , −8 ) ? - correct answer ✔✔a. No: For two ordered pairs to be
equal, the elements in each pair must occur in the same order. In this case the first element of
the first pair is 5, whereas the first element of the second pair is —5, and the second element of
the first pair is —5 whereas the second element of the second pair is 5.
b. Yes: Both pairs have the same elements 1212 and —8.
Let A = Z+, B = { n ∈ Z | 0 ≤ n ≤ 100}, and C = { 100, 200, 300, 400, 500 }. Evaluate the truth and
falsity of the following statement. - correct answer ✔✔True.
Every element in C is in C.
In general, the definition of subset implies that all sets are subsets of themselves.
A function F from A to B is a relation from A to B that satisfies the following two properties:
a. for every element x of A, there is ________________________.
b. for all elements x in A and y and z in B, if __________________________ then
__________________________. - correct answer ✔✔a. an element y of B such that ( x, y ) ∈ F
(i.e., such that x is related to y by F)
b. ( x, y ) ∈ F and ( x, z ) ∈ F; y = z
Define a relation from R to R as follows:
,For all ( x, y ) ∈ R × R, ( x, y ) ∈ L means that y=x−1.
Is L a function? If it is, find L( 0 ) and L( 1 ) - correct answer ✔✔L is a function. For each real
number x, y = x−1 is a real number, and so there is a real number y with ( x, y ) ∈ L. Also if ( x, y )
∈ L and ( x, z ) ∈ L, then y = x − 1 and z = x − 1, and so y = z.
So,
L(0) = 0 − 1 = −1 and
L(1) = 1 − 1 = 0.
You can also check these results by inspecting the graph of L, shown below. Note that for every
real number x, the vertical line through (x, 0) passes through the graph of L exactly once. This
indicates both that every real number x is the first element of an ordered pair in L and also that
no two distinct ordered pairs in L have the same first element.
Fill in the blanks below so that argument (b) has the same form as argument (a). Then represent
the common form of the arguments using letters to stand for component sentences.
( a ). If Jane is a math major or Jane is a computer science major, then Jane will take Math 150.
Jane is a computer science major.
Therefore, Jane will take Math 150.
( b ). If logic is easy or ________________ ,then _______________.
I will study hard.
Therefore, I will get an A in this course. - correct answer ✔✔I (will) study hard.
I will get an A in this course.
Common form:
If p or q, then r.
q.
Therefore, r.
,What is the following statement in the form "If A then B" and the contrapositive of the form?
George's attaining age 35 is a necessary condition for his being president of the United States. -
correct answer ✔✔If A then B: If George has not attained the age of 35, then he cannot be
president of the United States.
the contrapositive of the "If A then B" : If George can be president of the United States, then he
has attained the age of 35.
Write negation, contrapositive, converse and inverse for the following statement.
If P is a square, then P is a rectangle. - correct answer ✔✔Negation: P is a square and P is not a
rectangle.
Contrapositive : If P is not a rectangle, then P is not a square.
Converse : If P is a rectangle, then P is a square.
Inverse : If P is not a square, then P is not a rectangle.
p∨q
∼p
∴q
What is the Rule of Inference? - correct answer ✔✔Elimination
Use symbols to write the logical form of the following arguments. If the argument is valid,
identify the rule of inference that guarantees its validity. Otherwise, state whether the converse
or the inverse error is made.
a. If this computer program is correct, then it produces the correct output when run with the
test data my teacher gave me.
This computer program produces the correct output when run with the test data my teacher
gave me.
∴ This computer program is correct.
, b. Sandra knows Java and Sandra knows C++.
∴ Sandra knows C++. - correct answer ✔✔a. p ⟶⟶ q
q∴p
invalid, converse error
b. p ∧∧ q
∴q
valid, generalization
De Morgan's laws say (1) that the negation of an and statement is logically equivalent to the
____________ statement in which each component is ______________, and (2) that the
negation of an or statement is logically equivalent to the _________________ statement in
which each component is _____________ . - correct answer ✔✔or;
negated;
and;
negated
p∨q
p→r
q→r
∴r
What is the Rule of Inference? - correct answer ✔✔Proof by Division into Cases
Use symbols to write the logical form of the following argument. If the argument is valid,
identify the rule of inference that guarantees its validity. Otherwise, state whether the converse
or the inverse error is made.
If I get a Christmas bonus, I'll buy a stereo.
answers already passed
A universal existential statement is a statement - correct answer ✔✔that is universal because its
first part says that a certain property is true for all objects of a given type, and it is existential
because its second part asserts the existence of something.
a. Is ( 5, −5 ) = ( −5, 5 ) ?
b. Is ( −2−4−2−4 , (−2)3 ) = ( 3636 , −8 ) ? - correct answer ✔✔a. No: For two ordered pairs to be
equal, the elements in each pair must occur in the same order. In this case the first element of
the first pair is 5, whereas the first element of the second pair is —5, and the second element of
the first pair is —5 whereas the second element of the second pair is 5.
b. Yes: Both pairs have the same elements 1212 and —8.
Let A = Z+, B = { n ∈ Z | 0 ≤ n ≤ 100}, and C = { 100, 200, 300, 400, 500 }. Evaluate the truth and
falsity of the following statement. - correct answer ✔✔True.
Every element in C is in C.
In general, the definition of subset implies that all sets are subsets of themselves.
A function F from A to B is a relation from A to B that satisfies the following two properties:
a. for every element x of A, there is ________________________.
b. for all elements x in A and y and z in B, if __________________________ then
__________________________. - correct answer ✔✔a. an element y of B such that ( x, y ) ∈ F
(i.e., such that x is related to y by F)
b. ( x, y ) ∈ F and ( x, z ) ∈ F; y = z
Define a relation from R to R as follows:
,For all ( x, y ) ∈ R × R, ( x, y ) ∈ L means that y=x−1.
Is L a function? If it is, find L( 0 ) and L( 1 ) - correct answer ✔✔L is a function. For each real
number x, y = x−1 is a real number, and so there is a real number y with ( x, y ) ∈ L. Also if ( x, y )
∈ L and ( x, z ) ∈ L, then y = x − 1 and z = x − 1, and so y = z.
So,
L(0) = 0 − 1 = −1 and
L(1) = 1 − 1 = 0.
You can also check these results by inspecting the graph of L, shown below. Note that for every
real number x, the vertical line through (x, 0) passes through the graph of L exactly once. This
indicates both that every real number x is the first element of an ordered pair in L and also that
no two distinct ordered pairs in L have the same first element.
Fill in the blanks below so that argument (b) has the same form as argument (a). Then represent
the common form of the arguments using letters to stand for component sentences.
( a ). If Jane is a math major or Jane is a computer science major, then Jane will take Math 150.
Jane is a computer science major.
Therefore, Jane will take Math 150.
( b ). If logic is easy or ________________ ,then _______________.
I will study hard.
Therefore, I will get an A in this course. - correct answer ✔✔I (will) study hard.
I will get an A in this course.
Common form:
If p or q, then r.
q.
Therefore, r.
,What is the following statement in the form "If A then B" and the contrapositive of the form?
George's attaining age 35 is a necessary condition for his being president of the United States. -
correct answer ✔✔If A then B: If George has not attained the age of 35, then he cannot be
president of the United States.
the contrapositive of the "If A then B" : If George can be president of the United States, then he
has attained the age of 35.
Write negation, contrapositive, converse and inverse for the following statement.
If P is a square, then P is a rectangle. - correct answer ✔✔Negation: P is a square and P is not a
rectangle.
Contrapositive : If P is not a rectangle, then P is not a square.
Converse : If P is a rectangle, then P is a square.
Inverse : If P is not a square, then P is not a rectangle.
p∨q
∼p
∴q
What is the Rule of Inference? - correct answer ✔✔Elimination
Use symbols to write the logical form of the following arguments. If the argument is valid,
identify the rule of inference that guarantees its validity. Otherwise, state whether the converse
or the inverse error is made.
a. If this computer program is correct, then it produces the correct output when run with the
test data my teacher gave me.
This computer program produces the correct output when run with the test data my teacher
gave me.
∴ This computer program is correct.
, b. Sandra knows Java and Sandra knows C++.
∴ Sandra knows C++. - correct answer ✔✔a. p ⟶⟶ q
q∴p
invalid, converse error
b. p ∧∧ q
∴q
valid, generalization
De Morgan's laws say (1) that the negation of an and statement is logically equivalent to the
____________ statement in which each component is ______________, and (2) that the
negation of an or statement is logically equivalent to the _________________ statement in
which each component is _____________ . - correct answer ✔✔or;
negated;
and;
negated
p∨q
p→r
q→r
∴r
What is the Rule of Inference? - correct answer ✔✔Proof by Division into Cases
Use symbols to write the logical form of the following argument. If the argument is valid,
identify the rule of inference that guarantees its validity. Otherwise, state whether the converse
or the inverse error is made.
If I get a Christmas bonus, I'll buy a stereo.
