Archive for the ‘Number System’ Category
Problem 1008
Let p and q represent the following simple statements:
p : A basketball is round.
q : A hockey stick is round.
Write the following compound statement in its symbolic form.
A basketball is round and a hockey stick is not round.
Solution:
The given compound statement is a conjunction. A conjunction is a compound statement formed by joining two simple statement with the connective and.
“A basketball is round ” is the first simple statement in the conjunction; it preceded the connective and.
This simple statement is represented by the letter p.
“A hockey stick is not round” is the second simple statement in the conjunction; it follows the connective and.
Notice that this is the negation of the statement ”A hockey stick is round,” represented by q.
The symbolic form of the statement “A hockey stick is not round” is ̴q.
The symbol that represents the connective and is Λ .
To write the conjunction in symbolic form, replace each simple statement with its symbolic form. Then replace the connective and with the symbol that represent it.
A basketball is round and a hockey stick is not round.
p Λ ̴q
The symbolic form of the conjunction is p Λ ̴q
Problem 1007
a) Express the quantified statement in an equivalent way, that is, in a way that has exactly the same meaning.
b) Write the negation of the quantified statement .(The negation should begin with “all”, “some”, or “no.”)
Some babies are cute.
Solution:
a) Using our knowledge of the English language, we can express quantified statement in two ways that have exactly the same meaning. These equivalent are shown in the following table.
Statement  An Equivalent Way to Express the Statement 
All A are B.  There are no A that are not B. 
Some A are B.  At least one A is a B.

No A are B.  All A are not B.

Some A are not B.

Not all A are B. 
Using the table , the statement that is equivalent to “Some babies are cute” is “At least one baby is cute.”
b) Negations of quantified statements are summarized in the following table. (The negations of the statements in the second column are the statement in the first column.)
Statement  Negation 
All A are B.  Some A are not B. 
Some A are B.  No A are B. 
Using the table, the statement that is the negation of “Some babies are cute” is “No babies are cute.”
Problem 1006
The statement listed below is false. Let p represent the statement.
p: Goldfish are small pets.
Express the symbolic statement ̴p in words. What can be concluded about the resulting verbal statement?
a) In words, the symbolic statement ̴p is
b) What can you conclude about the resulting verbal statement?
Solution:
a) Goldfish are not small pets.
b) The resulting verbal statement is true because p is false.
Problem 1005
The statement listed below is false. Let p represent the statement.
p: The adult human skull contain of two bones.
Express the symbolic statement ̴p in words. What can be concluded about the resulting verbal statement?
Solution:
The symbol ̴ is translated as “not.” Therefore, ̴p represents the negation of the given statement, p.
The negation of a statement is statement that has a meaning that is opposite that of the original meaning.
So , in words, the symbolic statement ̴p is
The adult human skull does not consist of two bones.
The resulting verbal statement is the negation of the initial verbal statement.
Thus, the resulting verbal statement is true because the initial verbal statement is false.