Survey of Math - MAT 140
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Compound Statements and Connectives
There are several ways to compose a compound statement. The words AND and OR come to mind, but there are also,
IF...THEN... and IF AND ONLY IF .i f f / : Though NOT is not technically a compound idea, we could put it here to
heighten the idea that it is not a simple statement.
*Note: Mathematicians are notoriously LAZY, that is why they always replace words and complete sentences with a
single or combination of symbols. Logic is no different.
So instead of writing a simple statement over and over again we just let a letter represent the whole thing! For instance:
"There is a snake in my boot." could be replaced by the letter p
p "There is a snake in my boot"
Now everywhere that we see the symbolic p it represents the simple statement.
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NOT (negation) ~
The symbolic form of the word NOT is ~ . So if we wanted to say that:
"There is not a snake in my boot" we could replace this statement with: ~ p
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AND (conjunction) ^
The symbolic form of the word AND is ^: So if we wanted to represent with symbols:
"Mary is going to the bank AND to the Store" we would rst have to set up what the simple statements are and
p
Mary is going to the bank.
represent them by individual letters:
q
Mary is going to the store.
Now we can represent the entire statement: "Mary is going to the bank AND to the Store" with
p^q
If we wanted to represent the statement: "Mary is going to the bank AND NOT to the Store" we have:
p ^ ~q
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
3
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OR (disjunction) _
The symbolic form of the word OR is _: To use it to represent the statement:
"Shawn will play baseball OR Emily will not play kickball" we rst have to nd representations for our Simple
p
Shawn will play baseball
statements using letters:
notice that the word NOT is not in q
q
Emily will play kickball.
Now we can represent: "Shawn will play baseball OR Emily will not play kickball" by:
p _ ~q
*A little side note: the word "BUT" is the same as "AND" and the words "NEITHER.this...NOR that" can be
changed to "Not this AND Not that".
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IF...THEN... (conditionals) !
The symbolic form of the connective IF...Then is ! : Here we have to be careful because there are two separated
words represented by only one symbol. So when this symbol is used we have to remember that the word IF needs to
go to the front of the statement!
"If George does his homework then he will not fail his class"
statements:
p
q
Again we need to symbolized the simple
George does his homework
George will fail his class
Now to change the statement: "If George does his homework then he will not fail his class" into symbolic form we
get:
p ! ~q
Example 1 Using the above simple statements, what would the English interpretation be of:
~p ^ q
Well that would be: " George did not do his homework and he will fail his class ".
Example 2 Using the given simple statements...
p
q
r
The school is closed
The beach is open
Mary is running late.
write the English sentence symbolically represented by: q ^ .~ p _ r / :
*Note: Here there are three ideas, so it will be necessary to group them in English the way they are grouped in
mathematical jargon. The symbol that groups (separates) in English is the Comma ( , ). So:
" The beach is open , AND the school is not closed OR Mary is running late. "
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
Example 3 Using the given simple statements...
p
q
r
The school is closed
The beach is open
Mary is running late.
write the English sentence symbolically represented by: ~r ! .~q ^ p/
" IF Mary is not running late , then the beach is not open AND the school is closed. "
Example 4 Using the given simple statements...
p
q
George does his homework
George will fail his class
write the English interpretation of: ~ . p _ ~q/
*Note: When a statement in ./ is started with a negation, the English starts with
"it is not true" or " it is false that"
" It is not True that, George does his homework OR that he will not fail his class "
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If and Only If "iff" (biconditional) $
The symbolic form of If and Only If is the double arrow $. I have also seen
as a symbolic representation. Here
all the words are represented but they are located together so using it is like AND or OR.
Example 5 " It is raining outside iff there are clouds in the sky "
By now you know what is coming next, symbolic representation of the simple statements:
p
q
It is raining outside
There are clouds in the sky.
Thus symbolically we get:
p$q
Example 6 " It is false that, I will not get a degree if and only if I pass all my classes"
symbolic representation
p
q
I will get a degree
.
I will pass all my classes
Remember that "it is false that" must be a negation outside of a set of ./ .
~ ./
~ .~ p
! q/
Copyright 2007 by Tom Killoran