UNIT 1 MODULE 5

THE CONDITIONAL STATEMENT AND ITS VARIATIONS

 

THE CONDITIONAL STATEMENT

A conditional statement is a statement of the form"If p, then q."

The symbol for this "if...then" connective is the arrow: arrow

In other words, the statement "if p, then q" is denoted p arrow q

 

EXAMPLE 1.5.1

Let p represent "You drink Pepsi."

Let q represent "You are happy."

In this case p arrow q is the statement: "If you drink Pepsi, then you are happy."

 

Terminology:

"You drink Pepsi" is called the antecedent.

"You are happy" is called the consequent.

 

EXAMPLE 1.5.2

Let p be the statement "It rains."

Let q be the statement "I stay home."

Symbolize each statement.

1. If it rains, then I stay home.

2. It is not the case that if it rains, then I stay home.

3. If I don't stay home, then it doesn't rain.

4. It is not the case that if I stay home, then it doesn't rain.

 

Solution to Example 1.5.2

1. p arrow q

2. not (p arrow q)

3. not q arrow not p

4. not (q arrow not p)

 

 

VARIATIONS ON THE CONDITIONAL STATEMENT

Direct statement:
If p, then q.
p arrow q

Converse:
If q, then p.
q arrow p

Inverse:
If not p, then not q.
not p arrow not q

Contrapositive:
If not q, then not p.
not q arrow not p

 

Examples

Direct statement: If you drink Pepsi, then you are happy.

Converse: If you are happy, then you drink Pepsi.

Inverse: If you don't drink Pepsi, then you aren't happy.

Contrapositive: If you aren't happy, then you don't drink Pepsi.

 

 

EXAMPLE 1.5.3

Symbolize this statement, taken from the instructions for IRS From 1040, line 10:

If you received a refund of state income taxes or you received a refund of local income taxes, then, if your itemized deduction of state income taxes resulted in a tax benefit or your itemized deduction of local income taxes resulted in a tax benefit, then you must report this tax benefit as income.  

Let
p: you received a refund of state income taxes
q: you received a refund of local income taxes
r: your itemized deduction of state income taxes resulted in a tax benefit
s: your itemized deduction of local income taxes resulted in a tax benefit
w: you must report this tax benefit as income  


Solution to Example 1.5.3
(p or q) arrow [ (r or s) arrow w ]

 

 

WWW Note:
For practice problems involving translation of statements from words into symbols and vice-versa, visit Mr. Wooland's home page and try The Symbolizer

EXAMPLE 1.5.4

Let p be the statement "You drink Pepsi."

Let q be the statement "You are happy."

Make a truth table for the statement p arrow q.

 

 

 

See detailed explanation  

 

 

FUNDAMENTAL BEHAVIOR OF THE CONDITIONAL STATEMENT

The only situation in which a conditional statement is FALSE is when the ANTECEDENT is TRUE while the CONSEQUENT is FALSE.

 

 

EXAMPLE 1.5.5

1. Let p represent a true statement, while q and r represent false statements. Determine the truth value of this compound statement:

(p arrow not q) or r

 

 

2. Let p, s, and w represent true statements, while q, r and u represent false statements. Determine the truth value of this compound statement:

not p arrow {not [ (q or not r) and (w and not q) or (u and not w)] }

  Hint for problem #2: This particular problem is not as complicated as it at first appears to be.

 

See solutions
 

WWW Note:
For practice problems involving truth values of symbolic statements, visit Mr. Wooland's home page and try The Logicizer

EXAMPLE 1.5.6

Complete the following truth table.

table four rows and eight columns

see solution

Note: you will not be required to do truth table exercises.

 






WWW Note:
For practice problems involving truth tables, visit Mr. Wooland's home page and try The Truth Tabler

A FACT ABOUT EQUIVALENCY:

"If p, then q" is logically equivalent to "not p, or q"

Symbolically:



We can use a truth table to verify this claim.

see the truth table





EXAMPLE 1.5.7

Select that statement that is logically equivalent to: "If you don't carry an umbrella, you'll get soaked."

A. You carry an umbrella and you won't get soaked.

B. You carry an umbrella or you get soaked.

C. You don't carry an umbrella and you get soaked.

D. You don't carry an umbrella or you get soaked.

E. You leave your umbrella in the classroom, so you get soaked anyway.

see solution

 

THE NEGATION OF THE CONDITIONAL STATEMENT

Fact:

The negation of "if p, then q" is "p, and not q"

Symbolically:

not (p arrow q) equivalent to p and not q

We can use a truth table to verify this claim.

see the truth table

 



EXAMPLE 1.5.8

1. Select the statement that is the negation of "If you know the password, then you can get in."

A. If you don't know the password, then you can get in.

B. You don't know the password or you can get in.

C. You don't know the password and you can't get in.

D. You know the password and you can't get in.

 

2. Select the statement that is logically equivalent to "If you pass MGF1106, then a liberal studies math requirement is fulfilled."

A. If a liberal studies math requirement is fulfilled, then you passed MGF1106.

B. You pass MGF1106 and a liberal studies math requirement is fulfilled.

C. You don't pass MGF1106, or a liberal studies math requirement is fulfilled.

D. You pass MGF1106, or a liberal studies math requirement is not fulfilled.

 

 

3. Select the statement that is the negation of "If you have income from royalties, then you must complete Schedule E."

A. You have income from royalties and you must complete Schedule E.

B. You have income from royalties and you don't have to complete Schedule E.

C. You have income from royalties or you must complete Schedule E.

D. You have income from royalties or you don't have to complete Schedule E.

see solution

 

 

EXAMPLE 1.5.9

1. Select the statement that is the negation of "If we get a pay raise, then we will be content."

A. If we don't get a pay raise, then we won't be content.

B. We get a pay raise and we are content.

C. We get a pay raise and we aren't content.

D. We don't get a pay raise or we aren't content.

 

2. Select the statement that is logically equivalent to "If it is raining, then we will watch TV."

A. It isn't raining or we don't watch TV.

B. It isn't raining or we watch TV.

C. It is raining and we watch TV.

D. It is raining and we don't watch TV.

E. It is not safe to watch TV in the rain.

 

3. Select the statement that is the negation of "If a dog wags its tail, then it won't bite."

A. A dog wags its tail and it bites.

B. A dog wags its tail and it doesn't bite.

C. A dog doesn't wag its tail or it bites.

D. If a dog doesn't wag its tail, then it will bite.

 

 

See solutions

 

Some informal equivalencies for the conditional statement

"If p, then q" is equivalent to "All p are q."

"If p, then not q" is equivalent to "No p are q."

 

Example:

"If something is a poodle, then it is a dog" is a round-about way of saying "All poodles are dogs."

 

Likewise,

"If something is a dog, then it isn't a cat" means the same as "No dogs are cats."

 

 

Recall the following from our introduction to the conditional statement:

VARIATIONS ON THE CONDITIONAL STATEMENT

Direct statement:
If p, then q.
p arrow q

Converse:
If q, then p.
q arrow p

Inverse:
If not p, then not q.
not p arrow not q

Contrapositive:
If not q, then not p.
not q arrow not p

 

Examples

Direct statement: If you are a hound dog, then you howl at the moon.
Converse: If you howl at the moon, then you are a hound dog.
Inverse: If you aren't a hound dog, then you don't howl at the moon.
Contrapositive: If you don't howl at the moon, then you aren't a hound dog.
 

 

EXAMPLE 1.5.10

1. Select the statement that is the converse of "If I had a hammer, I would hammer in the morning."

A. If I didn't have a hammer, I wouldn't hammer in the morning."

B. If I don't hammer in the morning, I don't have a hammer.

C. If I hammer in the morning, I have a hammer.

D. If I had a ham, I would eat ham in the morning.

 

2. Select the statement that is the inverse of "If it rains, then I won't go to class."

A. If I don't go to class, then it rains.

B. If it doesn't rain, then I will go to class.

C. If I go to class, then it isn't raining.

D. Since it's Friday I probably won't go to class, anyway.

 

3. From Shakespeare (Henry IV, Part II): Select the statement that is the negation of this line, spoken by Falstaff addressing Doll Tearsheet: "If the cook help to make the gluttony, you help to make the diseases."

A. If the cook doesn't help to make the gluttony, you don't help to make the diseases.

B. If you help to make the diseases, the cook helps to make the gluttony.

C. If you don't help to make the diseases, the cook doesn't help to make the gluttony .

D. The cook helps to make the gluttony and you don't help to make the diseases.

 

FACTS ABOUT CONVERSE-INVERSE-CONTRAPOSITIVE

The direct statement is equivalent to the contrapositive.

p arrow q equivalent to not q arrow not p

 

The converse is equivalent to the inverse. q arrow p equivalent to not p arrow not q

 

These claims can be verified by using truth tables.

If you make a truth table having columns for all four statements listed above you will see that the column for p arrow q is identical to the column for npt q arrow not p, but these two columns are different from the column for q arrow p. However, the column for q arrow p will be identical to the column for not p arrow not q.

 

 

EXAMPLE 1.5.11

1. Select the statement that is logically equivalent to "If today is Sunday, then school is closed."

A. If today isn't Sunday, then school isn't closed.

B. If school is closed, then today is Sunday.

C. If school isn't closed, then today isn't Sunday.

D. A, B, & C are all equivalent to the statement above.

 

 

2. Select the statement that is logically equivalent to "If you are a duck, then you aren't willing to waltz." (Adapted from Lewis Carrol.)

A. If you aren't willing to waltz, then you are a duck.

B. If you aren't a duck, then you are willing to waltz.

C. If you are willing to waltz, then you aren't a duck.

D. A, B & C are all equivalent to the given statement.

 

 

3. Select the statement that is not equivalent to "If I don't invest wisely, then I'll lose my money."

A. I invest wisely or I lose my money.

B. If I don't lose my money, then I invested wisely.

C. I lose my money or I invest wisely.

D. If I invest wisely, then I won't lose my money.

See solutions  

 

 

WWW Note:
For practice problems involving negations, equivalencies, DeMorgan's Laws and variations on the conditional statement, visit Mr. Wooland's home page and try The Implicator

EXAMPLE 1.5.12

Select the statement that is logically equivalent to:
"If all of my friends got hired, then some losers are gainfully employed."

A. If some losers are not gainfully employed, then none of my friends got hired.
B. If no losers are gainfully employed, then some of my friends didn't get hired.
C. If some of my friends didn't get hired, then no losers are gainfully employed.
D. If none of my friends got hired, then some losers aren't gainfully employed.












SUMMARY

Here is a summary of facts about the conditional statement and its variations:






EXAMPLE 1.5.13

A passage from Lewis Carroll:
"And now, if e'er by chance I put
My fingers into glue,
Or madly squeeze a right-hand foot
Into a left-hand shoe,
Or if I drop upon my toe
A very heavy weight,
I weep,for it reminds me so
Of that old man I used to know ..."

Select the statement that is equivalent to "If I put my fingers into glue or squeeze a right-hand foot into a left-hand shoe or drop a heavy weight upon my toe, then I weep."

A. If I don't weep then I don't put my fingers into glue or squeeze a right-hand foot into a left-hand shoe or drop a heavy weight upon my toe.
B. I put my fingers into glue or squeeze a right-hand foot into a left-hand shoe or drop a heavy weight upon my toe and I don't weep.
C. I don't put my fingers into glue and I don't squeeze a right-hand foot into a left-hand shoe and I don't drop a heavy weight upon my toe, or I weep.
D. If I weep, then I put my fingers into glue or squeeze a right-hand foot into a left-hand shoe or drop a heavy weight upon my toe.











EXAMPLE 1.5.14 "I didn't know I was to have a party at all," said Alice; "but if there is to be one, I think I ought to invite the guests."
Select the statement that is the negation of "I didn't know I would have a party, but if I will have a party I will choose the guests."

A. I knew I would have a party and if I won't have a party then I won't choose the guests.
B. I knew I would have a party or, I will have a party but I won't choose the guests.
C. I knew I would have a party or if I won't have a party then I won't choose the guests.
D. I knew I would have a party and, I will have a party but I won't choose the guests.




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