# Lecture 3 notes Lecture 3 Winter 2009 notes including compound and converse statements

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MATH 135 Winter 2009

Lecture III Notes

Notation

Some notation to remember:

Z={. . . −3,−2,−1,0,1,2,3, . . .}= set of integers

P={1,2,3,4, . . .}= set of positive integers

N.B. This notation is not standard, but is used by our textbook.

R= the set of all real numbers

Compound Statements

If Aand Bare mathematical statements, we often see compound statements such as “Aand B” and

“Aor B”.

We deﬁne what the words “and” and “or” mean mathematically by using an organizational chart

called a truth table:

A B A and B A or B

T T T T

T F F T

F T F T

F F F F

Atruth table lists all possible combinations of TRUE (T) and FALSE (F) for the original statements

and tells us whether the compound statement is TRUE or FALSE in each case. (Truth tables can

be used to look at more complex compound statements, but we won’t do this in MATH 135.)

From the table, for “Aand B” to be TRUE, both Aand Bmust be TRUE.

Otherwise (when one is FALSE or both are FALSE), “Aand B” is FALSE.

For “Aor B” to be TRUE, either or both of Aand Bmust be TRUE.

Otherwise (when both are FALSE), “Aor B” is FALSE.

This makes sense when we consider the normal English language usage of these words.

Example

A=“2 is a prime number”, B=“5 is a perfect square”

Is “Aand B” TRUE or FALSE?

Is “Aor B” TRUE or FALSE?

Aside Regarding Sets

Recall that if Aand Bare sets, then A∪Bis the set of elements that are in either Aor

B, and A∩Bis the set of elements that are in both Aand B.

So A∪Bis similar to “Aor B”, and A∩Bis similar to “Aand B”.

Example

If A={1,2,4,5,6,9,10}and B={2,3,6,7,8}, then

A∪B={1,2,3,4,5,6,7,8,9,10}

A∩B={2,6}

Also, the statement NOT Ais called the negation of A. (With the example above, NOT Ais “2 is

not a prime number.) If statement Pis TRUE, then NOT Pis FALSE, and vice versa.

## Document Summary

This notation is not standard, but is used by our textbook. R = the set of all real numbers. If a and b are mathematical statements, we often see compound statements such as a and b and. We de ne what the words and and or mean mathematically by using an organizational chart called a truth table: From the table, for a and b to be true, both a and b must be true. Otherwise (when one is false or both are false), a and b is false. For a or b to be true, either or both of a and b must be true. Otherwise (when both are false), a or b is false. This makes sense when we consider the normal english language usage of these words. A = 2 is a prime number , b = 5 is a perfect square .