Writing Statements in If , then ' format and Counterexamples

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Writing Statements in If , then . format and
Counterexamples
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Consider the statement below
If a student takes an Internet course, then that
student uses a computer.
Which Venn Diagram correctly models the statement?
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If a student takes an Internet course, then that
student uses a computer.
The winner is!!
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If a student takes an Internet course, then that
student uses a computer.
Another way to state this is All students who
take an Internet Course use a computer.
Students who use a computer make up one set.
Students who take an Internet Course form a
subset of students who use a computer. A subset
is shown in a Venn diagram by placing its circle
inside the circle that it is a subset of.
Students who
Students who
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If a student takes an Internet course, then that
student uses a computer.
A little more explanation From the statement, if
a student takes an Internet course, you know that
the student uses a computer. So using a computer
is a prerequisite to taking an Internet course.
Students who take an Internet course fall under
the category of students who use a computer.
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Students who
Do not leave this page until it makes sense to
you that this Venn diagram models the if, then
statement!
Students who
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We have seen that the statements All students
who take an Internet Course use a computer.
and If a student takes an Internet course,
then that student uses a computer. are
logically equivalent.
Write an if, then statement that is equivalent
to All counting numbers divisible by four are
even. Include a Venn diagram. Note if the
word all is in your answer, then it is wrong!
The Counting Numbers are the set 1, 2, 3, 4,
5, . We must specify this because the
statement does not hold for the number zero.
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We have seen that the statements All students
who take an Internet Course use a computer.
and If a student takes an Internet course,
then that student uses a computer. are
logically equivalent.
Write an if, then statement that is equivalent
to All counting numbers divisible by four are
even. Include a Venn diagram. Note if the
word all is in your answer, then it is wrong!
If a counting number is divisible by four, then
it is even.
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An if, then statement has two parts
If a counting number is divisible by four,
then it is even.
hypothesis
conclusion
For the statement to be true the conclusion must
always follow from the hypothesis. In any
situation where the hypothesis is true, the
conclusion must also be true. Otherwise the
statement as a whole is false.
If , then statements are also known as
Conditional Statements.
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Write an if, then statement that is equivalent
to All counting numbers divisible by six are
divisible by three. Include a Venn diagram.
Counting Numbers
If a counting number is a multiple of six, then
it is a multiple of three.
Remember, if the word all appears in your if,
then statement, it is not correct.
Sometimes the universal set, in this case
Counting Numbers is included in the diagram.
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Avoid abbreviating too much in your Venn diagram.
It can cause confusion.
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Write an if, then statement that is equivalent
to All cats are mammals. Include a Venn
diagram.
If it is a cat, then it is a mammal.
Remember, if the word all appears in your if,
then statement, it is not correct.
Avoid abbreviating too much in your Venn diagram.
It can cause confusion.
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Write an if, then statement that is equivalent
to No counting number divisible by four is
odd. Include a Venn diagram. Make the sets in
your Venn diagram positive that is, do not use
the words not or no in describing the sets.
In your if, then statement, you can use the
word not but you cannot use the word no.
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Write an if, then statement that is equivalent
to No counting number divisible by four is
odd. Include a Venn diagram. Make the sets in
your Venn diagram positive that is, do not use
the words not or no in describing the sets.
If a counting number is divisible by four, then
it is not odd.
An equivalent statement is
If a counting number is odd, then it is not
divisible by four.
To check that you have the correct if, then
statement, see that your Venn diagram works with
both statements. Here the two sets in the Venn
diagram are said to be disjoint. This means that
they have no elements in common. In this case,
one cannot find an odd number that is divisible
by four, nor can you find a number divisible by
four which is odd.
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Equivalent statements No counting number
divisible by four is odd. If a counting number is
divisible by four, then it is not odd. If a
counting number is odd, then it is not divisible
by four.
The following would NOT be equivalent
statements If a counting number is not
divisible by four, then it is odd. If a counting
number is not odd, then it is divisible by four.
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Write an if, then statement that is equivalent
to No students like tests. Include a Venn
diagram.
If one is a student, then one does not like tests.
If one likes tests, then one is not a student.
Always make the sets in your Venn diagrams
positive.
Notice that the statements, If one is not a
student, then one likes tests and If one does
not like tests, one is a student are not
equivalent to the original.
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Make a Venn diagram to model this
statement. Some students study.
This statement cannot be put into if, then
format
B
We know from the statement that some students
study, however, we cannot determine from the
statement whether there are students who do not
study.
I
Students could actually be a subset of Those who
study. However, since the statement leaves open
the possibility that some students do not study,
we make the Venn diagram as above to allow for
that possibility.
Suppose Judy is a student. One cannot determine
from the statement above whether Judy studies or
does not study. She could belong in the
intersection (I) or she may belong in (B).
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If, then statements are either true or false.
Determine whether each of the following are true
or false.

• If a number is even, then it is divisible by 4.
• If a number is divisible by two and by six, then
  it is divisible by 12.
• If two numbers are odd, then their product is
  even.
• If one number is odd and another is even, then
  their sum is odd.

For a conditional statement to be true, whenever
the hypothesis (if part) is true, the conclusion
(then part) must also be true.
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If, then statements are either true or false.
Determine whether each of the following are true
or false.

• If a number is even, then it is divisible by 4.
• If a number is divisible by two and by six, then
  it is divisible by 12.
• If two numbers are odd, then their product is
  even.
• If one number is odd and another is even, then
  their sum is odd.

FALSE
FALSE
FALSE
TRUE
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We will take these one at a time.
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If a number is even, then it is divisible by 4.
Some even numbers are divisible by 4, such as, 4,
8, 12, 100. Other even numbers are NOT divisible
by 4, such as, 6, 10, 14, 98
For the above if, then statement to be true,
whenever a number is even, it must also be
divisible by 4. Remember that an equivalent
statement is, All even numbers are
divisible by 4.
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To show that an if, then statement is false,
one need only find one example for which it
doesnt work. This example is called a
counterexample.
I would like you to write your counterexamples as
sentences. For instance
6 is an even number however, 6 is not divisible
by 4.
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Conditional Statement
If a number is even, then it is divisible by 4.
hypothesis
conclusion
Counterexample
6 is an even number however, 6 is not divisible
by 4.
different conclusion
hypothesis
Notice how the counterexample keeps the
hypothesis (if part) of the conditional
statement, but returns a conclusion (then part)
that is contrary to the conclusion of the
conditional statement. The counterexample shows
that the conditional statement is false. It is a
proof that it is false.
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Find a counterexample to prove the following
conditional statement false.
If a number is divisible by two and by six, then
it is divisible by 12.
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Find a counterexample to prove the following
conditional statement false.
If a number is divisible by two and by six, then
it is divisible by 12.
One possibility

18 is divisible by two and 18 is divisible by
six,
however 18 is NOT divisible by 12.
Other choices of numbers to use include 30, 42,
54, 66, Ever other multiple of 6, beginning
with 6, will do as a counterexample.
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Find a counterexample to prove the following
conditional statement false.

• If two numbers are odd, then their product is
  even.

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Here we need two numbers. One possibility

5 and 9 are two odd numbers however, their
product, 45, is not even.
Any two odd numbers will do as our
counterexample. This is because the product of
two odd numbers is actually always odd.
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The last conditional statement is true.
If one number is odd and another is even, then
their sum is odd.
Not matter what pair of numbers, one odd and one
even, that you chose this statement will hold.
However, you will find, that proving that a
conditional statement is true is much more
difficult than proving that a conditional
statement is false. No matter how many examples
you provide to show that the above statement
works, you still will not have proven that it
is true. This is because there may yet be
another example out there that doesnt work.
To prove that it is true, you will need to
develop a deductive argument. We will return to
the above statement in the activity, Reasoning in
the Classroom.
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You exercises will provide more practice
converting statements into if, then format and
producing counterexamples for false conditional
statements. Remember that for a conditional
statement to be true, it must always work. To
prove a conditional statement false, one need
only provide one instance where it does not
work.
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