Exam 1 Flashcards
What is a mathematical proposition?
A mathematical proposition is a statement which is either true or false.
What is the logical negation operation?
The logical negation operation reverses the truth value of a proposition.
What is a predicate?
A predicate is a statement whose truth values depend on one or more variables.
Let P and Q be propositions. When is the proposition “P and Q” true?
The proposition “P and Q” is true when both P and Q are true.
Let P and Q be propositions. When is the proposition “P or Q” false?
The proposition “P or Q” is false when both P and Q are false.
Let P and Q be propositions. When is the proposition “P implies Q” false?
The proposition “P implies Q” is false when P is true and Q is false.
Let P and Q be propositions. What is the converse of “P implies Q”?
The converse of “P implies Q” is “Q implies P”.
Let P and Q be propositions. What is the contrapositive of “P implies Q”?
The contrapositive of “P implies Q” is “not Q implies not P”.
Let P and Q be propositions. What is the proposition of “P if and only if Q” false?
The proposition “P if and only if Q” is false when P and Q have different truth values.
Give the definition of an even integer.
An integer nis even if n= 2k for some integer k.
Give the definition of an odd integer.
An integer n is odd if n= 2k+ 1 for some integer k.
Give the definition of a rational number.
A number r is rational if there exist two integers x and y, where y is nonzero, such that r= x/y.
What does it mean for an integer x to divide another integer y?
x divides y if there exists an integer k such that y= kx.
Give the definition of a prime number.
A prime number is an integer strictly greater than one which is only divisible by one and itself.
Give the definition of an irrational number.
An irrational number is a real number which is not rational.
Give the definition of a set.
A set is a collection of objects.
Give the definition of the empty set.
The empty set is the set with no elements.
Give the definition of a finite set.
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
Give the definition of the cardinality of a finite set A.
the cardinality of A is the number of elements in A.
Let A and B be sets. Give the definition of A being a subset of B.
A is a subset of B if every element of A is also an element of B.
Give the definition of the power set of a set A
The power set of A is the set of all subsets of A.
Let A and B be sets. Give the intersection of A and B.
The intersection of A and B is the set of all elements that are in both A and B.
Let A and B be sets. Give the union of A and B.
The union of A and B is the set of all elements that are in A or in B.
Let A be a set and let U be a universal set. Give the definition of the complement of A.
The complement of A is the set of all elements of the universal set which are not in A.
Let A and B be sets. Give the definition of the Cartesian product A ×B.
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.
Give the definition of two sets being disjoint.
Two sets are disjoint when their intersection is empty.
Let X and Y be sets. Give the definition of a function f from X to Y
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 .
Give the definition of the range of a function f : X →Y .
The range of f is the set {y ∈Y : there exists an x ∈X such that f (x) = y}.
Give the definition of two functions f : X →Y and g : X →Y being equal.
Two functions f : X →Y and g : X →Y are equal if, for every x ∈X, f (x) = g(x).
Let f : X →Y be a function. Give the definition of f being an injection.
f is an injection if, for every x1, x2 ∈X, if x1 ̸= x2 then f (x1) ̸= f (x2).
Let f : X →Y be a function. Give the definition of f begin a surjection.
f is a surjection if, for every y ∈Y , there exists x ∈X such that f (x) = y.
Give the definition of a bijection.
A bijection is a function which is both an injection and a surjection.
Let f : X →Y and g : Y →Z be functions. Give the definition of the composition g ◦f .
The composition g ◦f is the function from X to Z defined by (g ◦f)(x) = g(f (x)) for every x ∈X.
Give the definition of the identity function on a set X
The identity function is the function f from X to X which is defined by f (x) = x.
Let f : X →Y be a function. Give the definition of the inverse of f
Solution: The inverse of f is the function g : Y →X which satisfies f ◦g = idY and g ◦f = idX.
