Archive for the ‘Boolean algebra’ Category

Problem 1063

For the statement below, write the statement symbolically using parentheses and determine whether that statement is a negation, conjunction, dis-junction, conditional, or bi-conditional.

Statement:  If I don’t have to pay or I have enough money, then I can eat in the restaurant.

 

Solution:-

 

a. First break the compound statement down into its simple statements.

p: I have to pay.

q: I have enough money.

r: I can eat at the restaurant.

 

The comma indicates that the preceding portion of the compound statement should be placed in parentheses.

The word “or” indicates that the symbol v will be used.

The words “if” and “then” indicate that this is a conditional statement with r as the consequent and that the symbol → will be used.

The statement “I don’t have to pay” is the negative of the simple statement “I have to pay ” and is thus represented symbolically as ~p.

The “or” connective is in between the ~p and q statements, and that is represented as ~p v q.

If you combine all of these, you obtain (~p v q)→ r.

 

b.To determine whether the statement (~p v q) → r is a negation, conjunction, dis-junction, conditional, or bi-conditional, look at the symbol outside of the parentheses.

The → connective is outside the parentheses, so this indicates that the compound statement is a conditional.

 

 

 

 

Problem 1062

For the statement below, perform the following.

a. Write the statement symbolically by using parentheses.

b. Determine whether the statement is a negation is a negation, conjunction, dis-junction, conditional, or bi-conditional.

Statement: It is false that if your car is broken then you will not be late for work.

 

Solution:-

a. First break the compound statement down into its simple statements.

p: Your car is broken.

q: You will be late for work.

Recall that the symbol ‘→’ means “if-then” while the symbol ‘↔’ means “if and only if”. The symbol ‘~’ is used to negate a statement.

Notice that the statement is an “if-then” statement, the second simple statement is negate, and “it is false” represents a negation of the compound statement. Thus, the statement written symbolically is ~(p→~q).

 

b. Determine how to classify the given statement. Recall that the symbol ‘~’  means negation, the conditional is symbolized by ‘→’ , the bi-conditional is  symbolized by ‘↔’ , the bi-conditional is symbolized by ‘↔’, the symbol ‘Λ’ represents conjunctions, and ‘v’ represents dis-junctions. Look for the most dominant connective when trying to determine what kind of statement it is

Therefore , the statement found in part (a) is a negation.

 

 

 

Problem 1061

Let p, q, and r represent the following simple statements.

p: It is above 80 degrees.

q: It is raining.

r: I go swimming.

Write the symbolic statement in words.

(p v q) Λ~ r

 

Solution:-

 

First notice that this statement contains more than one connective. The connectives in this statement are “or” and “and.”

When a compound statement includes three simple statements and two of the simple statements are grouped using parentheses, a comma is used to separate the grouped statements from the other statement.

The compound statement “It is above 80 degrees or it is raining” is written symbolically as (p v q).

The last past of the statement is ~r. This represents “I do not go swimming.”

Thus, the symbolic statement (p v q) Λ ~ r means “It is above 80 degrees or it is raining, and I do not go swimming.”

 

Problem 1057

Write the statement in symbolic form. Let p and q represent the following statements.

p: The lion is fast.

q: The wildebeest is slow.

It is false that the lion is fast or the wildebeest is slow.

 

Solution:-

 

Notice that the phrase “it is false that” represents a negation that applies to the compound statement “the lion is fast or the wildebeest is slow.” When using symbols, place parentheses around the symbols that represent “ the lion is fast or the wildebeest is slow.”

Determine the proper symbol for each portion of the statement. Since the ~symbol represents negation, it is the symbol used for the first part of the statement.

It is false that   the lion is fast   or   the wildebeest is slow

~        (                                                                  )

Notice that p represents “the lion is fast.”

It is false that            the lion is fast               or      the wildebeest is slow

~                 (                  p                                              )

The symbol  v means “or.”

It is false that            the lion is fast               or      the wildebeest is slow

~                 (                            P                           v                           )

Finally, q represents “the wildebeest is slow.”

 

It is false that            the lion is fast     or      the wildebeest is slow

~                 (                  p                   v      q                )

Notice that the statement “It is false that” applies to the entire statement “the lion is fast or the wildebeest is slow.” Therefore , the symbolic form of the statement is  ~(p v q).

Problem 1056

Let p  and q represent the following simple statements.

p: It is Independence Day.

q: It is July 4th.

Write the following compound statement in its symbolic form.

It is not Independence Day if and only if it not July 4th.

 

Solution:-

 

The given compound statement is a biconditional statement.

A biconditional statement is a compound statement formed by joining two simple statements with the connective if and only if.

“It is not Independence Day” is the first simple statement in the biconditional; it precedes the connective if and only if.

Since this statement is the negation of the statement represented by p, its symbolic form is ~p.

“It is not July 4th” is the second simple statement in the biconditional; it follows the connective if and only if.

Since this statement is the negation of the statement represented by q, its symbolic form is ~q.

The symbol that represents the connective if and only if is ↔.

To write the conditional statement in symbolic form, begin by replacing the first simple statement with its symbolic form. Then, replace the connective if and only if by the symbol that represents it. Finally, write the second statement in symbolic form.

It is not Independence Day   if and only if    it is not July 4th.

~p                                  ↔              ~q

The symbolic form for the given biconditional statement is ~p ↔~q.

 

Problem 1055

Let p and q represent the following simple statements.

p: I jog.

q: I go to the gym.

Write the following compound statement in symbolic form.

I jog or I do not go to the gym.

 

Solution:-

 

The given compound statement is disjunction formed by joining two simple statement with the connective “or”.

Examine the first simple statement, “I jog.” This is the same as p in the problem statement.

Thus, “I jog” corresponds to p.

The connective “or” is represented by the symbol  v.

Now examine the second simple statement, “I do not go to the gym.” This is the negation of q in the problem statement . Thus ,” I do not go to the gym” corresponds to ~q.

Therefore , the statement “I jog or I do not go to the gym” corresponds to the symbolic form p v ~ q.

 

 

 

 

Problem 1054

Let p and q represent the following simple statements.

P: You eat your vegetables.

q: You get dessert.

Write the following compound statement in its symbolic form.

If you do not eat your vegetables, than you do not get dessert.

 

Solution:-

 

The given compound statement is a conditional statement.

A conditional statement is a compound statement formed by joining two statements with the connective if-then.

“You do not eat your vegetables” is the first simple statement in the conditional; it follows the word if.

Since this statement is the negation of the statement represented by p, its symbolic form is ~p.

“You do not get dessert” is the second simple statement in the conditional; it follows the word then.

Since this statement is the negation of the statement represented by q, its symbolic form is ~q.

The symbol that represents the connective if-then is →.

Note symbol the words “if” and “then” do not have separate symbols. A single symbol represents the connective if-then.

To write the conditional statement in symbolic form, begin by replacing the first simple statement with its symbolic form. Then, replace the connective if-then by the symbol that represents it. Finally, write the second simple statement in symbolic form.

If you do not eat your vegetables,   then  you do not get dessert.

~p                         →                         ~q

The symbolic form for the given conditional statement is

~p→~q.

 

 

Problem 1053

Write the statement is symbolic form.

Let

p: The emergency room is full

q: The patient is sick.

The patient is not sick or the emergency room is not full.

 

Solution:-

 

Notice that p represents the simple statement ‘The emergency room is full’ and q represents the simple statement ‘The patient is sick.’

Since the symbol “~” represents the negation of a statement, “the patient is not sick” is represented by ~q.

The patient is not sick     or the emergency room is not full

~q

To represent the “or” part of the statement, the symbol “v” is used.

To patient is not sick   or  the emergency room is not full.

~q               v

In the last part of the statement, ~p is used since p is being negated.

The patient is not sick    or   the emergency room is not full.

~q                v        ~p

Thus, the statement that “The patient is not sick or the emergency room is not full” is represented by ~q v ~p.

 

 

 

Problem 1052

Let p and q represent the following statements.

p: Brad Pitt is not governor.

q: Jimi Hendrix is still living.

Express the following statement symbolically.

Jimi Hendrix is not still living.

 

Solution:-

 

Symbolically, the statement is ~q.

 

 

Problem 1051

Let p and q represent the following statements.

p:  There are 33 books in the New Testament.

q: H2O is the chemical symbol for carbon dioxide.

Express the following statement symbolically.

There are not 33 books in the New Testament.

 

Solution:-

 

The negation of a statement is a statement that has a meaning that is opposite that of the original meaning.

The statement is the negation of the statement p.

The negation of a statement is expressed by writing the symbol “ ~ “ before the letter representing the statement. Therefore, the statement is expressed symbolically as ~p.