Exam 1

Question 1

What is a mathematical proposition?

Answer 1

A mathematical proposition is a statement which is either true or false.

Question 2

What is the logical negation operation?

Answer 2

The logical negation operation reverses the truth value of a proposition.

Question 3

What is a predicate?

Answer 3

A predicate is a statement whose truth values depend on one or more variables.

Question 4

Let P and Q be propositions. When is the proposition “P and Q” true?

Answer 4

The proposition “P and Q” is true when both P and Q are true.

Question 5

Let P and Q be propositions. When is the proposition “P or Q” false?

Answer 5

The proposition “P or Q” is false when both P and Q are false.

Question 6

Let P and Q be propositions. When is the proposition “P implies Q” false?

Answer 6

The proposition “P implies Q” is false when P is true and Q is false.

Question 7

Let P and Q be propositions. What is the converse of “P implies Q”?

Answer 7

The converse of “P implies Q” is “Q implies P”.

Question 8

Let P and Q be propositions. What is the contrapositive of “P implies Q”?

Answer 8

The contrapositive of “P implies Q” is “not Q implies not P”.

Question 9

Let P and Q be propositions. What is the proposition of “P if and only if Q” false?

Answer 9

The proposition “P if and only if Q” is false when P and Q have different truth values.

Question 10

Give the definition of an even integer.

Answer 10

An integer nis even if n= 2k for some integer k.

Question 11

Give the definition of an odd integer.

Answer 11

An integer n is odd if n= 2k+ 1 for some integer k.

Question 12

Give the definition of a rational number.

Answer 12

A number r is rational if there exist two integers x and y, where y is nonzero, such that r= x/y.

Question 13

What does it mean for an integer x to divide another integer y?

Answer 13

x divides y if there exists an integer k such that y= kx.

Question 14

Give the definition of a prime number.

Answer 14

A prime number is an integer strictly greater than one which is only divisible by one and itself.

Question 15

Give the definition of an irrational number.

Answer 15

An irrational number is a real number which is not rational.

Question 16

Give the definition of a set.

Answer 16

A set is a collection of objects.

Question 17

Give the definition of the empty set.

Answer 17

The empty set is the set with no elements.

Question 18

Give the definition of a finite set.

Answer 18

A finite set is a set which is either the empty set or which can be numbered 1 through n for some positive integer n

Question 19

Give the definition of the cardinality of a finite set A.

Answer 19

the cardinality of A is the number of elements in A.

Question 20

Let A and B be sets. Give the definition of A being a subset of B.

Answer 20

A is a subset of B if every element of A is also an element of B.

Question 21

Give the definition of the power set of a set A

Answer 21

The power set of A is the set of all subsets of A.

Question 22

Let A and B be sets. Give the intersection of A and B.

Answer 22

The intersection of A and B is the set of all elements that are in both A and B.

Question 23

Let A and B be sets. Give the union of A and B.

Answer 23

The union of A and B is the set of all elements that are in A or in B.

Question 24

Let A be a set and let U be a universal set. Give the definition of the complement of A.

Answer 24

The complement of A is the set of all elements of the universal set which are not in A.

Question 25

Let A and B be sets. Give the definition of the Cartesian product A ×B.

Answer 25

The Cartesian product of A and B is the set of all ordered pairs in which the first entry is in A and the second entry is in B.

Question 26

Give the definition of two sets being disjoint.

Answer 26

Two sets are disjoint when their intersection is empty.

Question 27

Let X and Y be sets. Give the definition of a function f from X to Y

Answer 27

A function f from X to Y is a subset of X ×Y such that, for every x ∈ X, there is a unique y ∈Y such that (x, y) ∈f .

Question 28

Give the definition of the range of a function f : X →Y .

Answer 28

The range of f is the set {y ∈Y : there exists an x ∈X such that f (x) = y}.

Question 29

Give the definition of two functions f : X →Y and g : X →Y being equal.

Answer 29

Two functions f : X →Y and g : X →Y are equal if, for every x ∈X, f (x) = g(x).

Question 30

Let f : X →Y be a function. Give the definition of f being an injection.

Answer 30

f is an injection if, for every x1, x2 ∈X, if x1 ̸= x2 then f (x1) ̸= f (x2).

Question 31

Let f : X →Y be a function. Give the definition of f begin a surjection.

Answer 31

f is a surjection if, for every y ∈Y , there exists x ∈X such that f (x) = y.

Question 32

Give the definition of a bijection.

Answer 32

A bijection is a function which is both an injection and a surjection.

Question 33

Let f : X →Y and g : Y →Z be functions. Give the definition of the composition g ◦f .

Answer 33

The composition g ◦f is the function from X to Z defined by (g ◦f)(x) = g(f (x)) for every x ∈X.

Question 34

Give the definition of the identity function on a set X

Answer 34

The identity function is the function f from X to X which is defined by f (x) = x.

Question 35

Let f : X →Y be a function. Give the definition of the inverse of f

Answer 35

Solution: The inverse of f is the function g : Y →X which satisfies f ◦g = idY and g ◦f = idX.