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.”