Archive for September, 2015
Problem 1015
Find the cardinal number for the following set A.
{x  x is a letter in the words isogram}
Solution:
The number of distinct, or different, element in set A is called the cardinal number of set A and is denoted n(A).
The elements of set A are letters that are in the word isograms.
Note that there are no repeated letters in the word isograms. Each letter is used exactly once. Therefore, the set A is equal A = {I, s, o, g, r, a, m}. If the letters were listed in alphabetical order, A = {a, g, I, m, o, r, s}. The order does not matter.
Count the number of element in set A to determine n(A).
n(A) = 7
Thus, the cardinal number of set A is 7.
Problem 1014
Write a description of the set.
{Africa, Antarctica, Asia, Australia}
Solution:
To given a description of a set is to classify all of the elements, or members, of the set using words. The description should be specific to only that set.
One possible description of the set is the set continents that begin with A.
Problem 1013
Determine if the collection is not well define and therefore is not a set.
Solution:
A set is a collection of object whose contents can be clearly determined. The object in a set are called the element, or members, of the set.
Since high schools in New York are building the can be clearly determined, the collection is well defined. The collection forms a set.
Problem 1012
Find the reference angle for the following angle.
142°
Solution:
A reference angle for the angle θ is the positive acute angle made by the terminal side of angle θ and the xaxis.
The reference angle of a negative angle is found by first finding its coterminal angle that lies between 0° and 360° and then finding the reference angle of that angle.
First, the coterminal angle for 142° lying between 0° and 360° cab be found by adding 360°.
142° +360° = 218°
The terminal side of the angle 218° lies in quadrant III since 180° < 218° < 270°.
Since the angle is in quadrant III, the angle formed by the xaxis and the specified angle θ is θ – 180°.
Calculate the angle, θ’ , formed by the xaxis and the specified angle θ.
Θ’ = 218° – 180°
= 38°
Therefore, the reference angle for – 142° is 38°.
Problem 1011
Find a solution for the equation. Assume that all angles in which an unknown appears are acute angles.
cos (2α+30°) = sin(3α – 60°)
Solution:
Since cos θ and sin θ are cofunction, they are related by an identity.
The identity is cos θ = sin (90° – θ).
Thus, for acute angles, if a function and the cofunction values are equal, the sum of the angles must be 90°.
2α + 30° + 3α – 60° = 90°
5α – 30° = 90°
5α = 120°
α = 24°
Problem 1010
Rewrite cot (81°) in terms of its co function.
Solution:
Each trigonometric function has a cofunction. The sine’s cofunction is the cosine, the tangent’s cofunction is the cotangent and the secant’s cofunction is the cosecant. Thus, the abbreviation of the cofunction of cot θ is tan θ.
According to the cofunction identities, the trigonometric function of any angle is equal to the cofunction of the angle’s complement.
Therefore, cot (81°) = tan (9)°.
Problem 1009
Let p, q, and r represent the following simple statements:
p: I lie on the sofa.
q: I take a nap.
r: I go jogging.
Write the following compound statement in its symbolic form.
I lie on the sofa and I take a nap, or I go jogging.
Solution:
First, notice that this statement contains more than one connective.
I lie on the sofa and I take a nap, or I go jogging.
When compound statement containing more than one connective are expressed in words, commas are used to indicate which simple statement are to be grouped together.
Two simple statement that appear on the same side of a comma are grouped together in parentheses when the statement is written symbolically.
Since the compound statement.
I lie on the sofa and I take a nap appears to the left of the comma, the symbolic statement representing it is grouped within parentheses.
The symbolic statement expressing
I lie on the sofa and I take a nap
Is (p Λ q).
Next , write the symbol that represents the connective. Then symbolically write the statement the follows the connective.
I lie on the sofa and I take a nap, or I go jogging.
(p Λ q) V r
The symbolic form of the compound statement
I lie on the sofa and I take a nap, or I go jogging.
Is (p Λ q) V r.
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.