§3.2: Compound Statements and Connectives MGF 1106-Peace
Def : A simple statement is one that conveys one idea with no connecting words. These are what we used in §3.1.
Def : Connecting words , also called connectives , are words that join simple statements together. They are and, or, if…then, and if and only if.
Def: Statements formed by combining two or more simple statements are called compound statements.
Ex. It is after 9:45AM and the teacher is still lecturing.
-Notice the two simple statements “It is after 9:45AM” “The teacher is still lecturing” are combined with the connective “and.”
We will study the four connective words in this section to determine their meaning and their symbolic form.
I.
And Statements (Conjunction)
-If p and q represent two simple statements, then the compound statement “p and q” is represented by p
∧ q
-Remember from §2.5, the word “but” also means “and” in math. So if you see the word “but” you know this is a conjunction. Other ways to write conjunction are “yet” and “nevertheless.” See table 3.3. From a Chapter 2 viewpoint
“And” represents intersection.
Ex.
Let p: It is snowing outside. Let q: I am cold.
Write the symbolic form of “It is snowing outside, but I am not cold.”
Answer: p ^ ~q
In these sections make sure you can go from English to symbolic and symbolic to English.
II.
Or Statements (Disjunction)
If I say “I took math or English this semester.” What do you think that means?__
You took either math or English___
In our everyday life, “or” tends to mean either one but not both. This is called the “ exclusive or.”
However in math, when using the word “or” we use the “inclusive or ” which means either one or BOTH.
So now what do you think “I took math or English this semester” would mean?_____________________
It would mean you took either math, or English, or both classes.
The symbolic form of “p or q” is p
∨ q .
Ex. Let p: I study, let q: I pass the course. Give the English version of p
∨
~ q .
Answer: I study or I do not pass the course.
III.
If…Then statements (Conditional)
Sometimes quantified statements can be written in easier to understand ways.
For example, all alligators are reptiles means if something is an alligator then it has to be a reptile.
Symbolically, we represent “If p, then q” as p
→ q .
Now there are MANY different ways to represent p
→ q in English. Just glancing at Table 3.4 we can see this.
Please don’t memorize this table. Just learn the basic “if p then q” English way of saying it. Have the table handy when you do your HW just in case something else pops up. I will not give you one of the other ways to represent if p then q on the test.
Ex.
Give the symbolic form of “If this is not a reptile, then this is not an alligator.”
First make your simple statement definitions. Let p: it is a reptile. Let q: it is an alligator.
Then in symbolic form we have: p
→
~ q

IV.
If and Only If Statements (Biconditional)
-Now if you noticed before, if you switch the order of a conditional statement, the truth of the statement might change.
Ex.
Does “If something is an alligator, then it is a reptile” mean the same thing as “If something is a reptile, then it is an alligator?” Absolutely not. Order is VERY IMPORTANT for the conditional.
-However, sometimes we have if statements where order does not matter.
For example: “If he is your husband then he is your spouse.”
-Ask yourself if you switched the order on that statement, would the meaning change? In this case it wouldn’t. For these special cases we use the biconditional: if and only if statements.
Symbolically “p if and only if q” is represented as p
↔ q .
-Once again Table 3.6 gives an obnoxiously large amount of ways to represent the biconditional. Don’t worry about memorizing this. Just recognize the basic form.
Ex.
Let p: you are human q: you have feathers. Give the English form of ~ p
↔ q
Notice the negation on p before you begin. The English version would be “You are not human if and only if you have feathers.”
V.
Dominance of Connectives and the Order is which we Perform Connectives.
-Similar to order of operations in that we do all operations in parentheses first!
-Different from order of operations because instead of performing the most dominant connective first, we group all things before and after the most dominant connective and then perform other operations (sort of backwards).
Dominance of Connectives (from greatest dominance to least)
1.
Biconditional 2. Conditional 3. Conjunction (and), Disjunction (or) 4. Negation
Ex.
Take the following symbolic statement (which has no parentheses) and add parentheses to follow dominance of connectives. Then, translate to English. p: The temperature outside is above 85 degrees. q: We finished studying. r: We go to the beach. r
↔ q
∧ p
Answer: So what is the most dominant connective? The biconditional
So the rule is to group the items before the
↔ and after the
↔
↔
. together. We’ll use parentheses.
( r )
↔
( q
∧ p )
. We actually don’t need the parentheses around r since there is only item prior to the biconditional, but I put them for effect. Another way to write it as r
↔
( q
∧ p )
.
In English: We go to the beach if and only if we finished studying and the temperature outside is above 85 degrees.
We didn’t need to put commas in the above statement, since we already had the dominance of connectives.
Ex: Take the following English statement and convert it to symbolic form. Note: Commas in English are like parentheses in math. If you see them, they are a form of grouping. However, if no commas appear, you need to use dominance of connectives to properly order the operations. a.
The lines go down, or if the transformer blows then we do not have power.
Answer: Let p: the lines go down, q: the transformer blows, r: we have power

Now go left to right (remember a comma means to group things before and after: ( p )
∨
( q
→
~ r )
Again the parentheses around the p are unnecessary: p
∨
( q
→
~ r ) . Now within the parentheses, we need to use dominance of connectives. What is the most dominant connective? The conditional and after the
→
: p
∨
( q
→
(~ r ))
→
. So we’ll group things before
[I didn’t put any parentheses around q since it was by itself and didn’t need it].
Generally if an item is isolated (no negations or other operations, we don’t need parentheses).
Do you see that if the comma had not been there, we would have not arranged our parentheses this way? b.
If the lines do not go down and we have power then the transformer does not blow or there is an increase in the cost of electricity. {Holy…..is this supposed to make sense??
☺
}
Answer: Let p: the lines go down, q: the transformer blows, r: we have power, s: increase in cost of electricity
There are no commas in here, so we will first write our expression symbolically from left to right without parentheses. Then we’ll go back and add them in for clarity.
~ p
∧ r
→
~ q
∨ s
Technically we could stop here, since dominance of connectives is implied. But let’s add parentheses to support dominance of connectives. What is the most dominant connective? The conditional
→
. So we’ll put parentheses before and after it: (~ p
∧ r )
→
(~ q
∨ s ) . Continuing as in the last example we have:
((~ p )
∧ r )
→
((~ q )
∨ s )