Translate these statements into English, where the domain
for each variable consists of all real numbers.
a
b
c
a. For every there exists a where is less than
b. For every and every if is positive and is positive then the product of and will be positive.
c. For all real numbers and there exists a number that is the product of and .
Let be the statement “x has sent an e-mail message
to y,” where the domain for both x and y consists of
all students in your class. Express each of these quantifications
in English.
a)
b)
c)
d)
e)
f)
a) There exists a student x who has emailed another student y
b) There exists a student x who has emailed every other student
c) All students have emailed at least one student in class.
d) There exists a student y who has received an email from everybody
e) Every student has received an email from student X
f) All students have sent and received an email from eachother
Express each of these statements using mathematical and
logical operators, predicates, and quantifiers, where the
domain consists of all integers.
a) The sum of two negative integers is negative.
b) The difference of two positive integers is not necessarily
positive.
c) The sum of the squares of two integers is greater than
or equal to the square of their sum.
d) The absolute value of the product of two integers is
the product of their absolute values.
a)
b)
Book answer is
c)
d)
Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that every positive integer is the sum of the squares of four integers.
Book answer
Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.
a) The product of two negative real numbers is positive.
b) The difference of a real number and itself is zero.
c) Every positive real number has exactly two square
roots.
d) A negative real number does not have a square root
that is a real number.
a)
b)
c)
We must have two existentially quantified variables to represent the two square roots. We must specify that the two objects are different. We must say that an object meets the conditions iff it is one of those two.
("For all there exists an and a where and are different and for all , is if and only iff and ")
d)