RSN – Reasoning Notes
Mr. Erlin – Geometry
RSN01 Given a conditional (If, Then) statement, identify the hypothesis and conclusion.
Notes: Whatever follows the “IF” is the hypothesis. Whatever follows the “THEN” is the
conclusion. Be sure to NOT include the words IF and THEN in the Hypothesis or
Conclusion…those are just indictors of what is about to come…not part of the result.
We will be using P to stand in as the Hypothesis and Q to stand in as the Conclusion in our later
lessons. This will save us time writing and make the actual “manipulations” more evident rather
than getting us lost in the context of the specific Conditional statement.
Also note that we can use an arrow to indicate IF…THEN. It must be placed after the Hypothesis
and pointing towards the Conclusion. A double sided arrow will come to mean something
else…so direction is very important.
RSN02
Rewrite a statement in appropriate If Then conditional form.
When a statement has a conditional relationship between two events, it can be written as a
Conditional Statement. The trick is to think about Euler (Oiler) Diagrams. The Hypothesis is
always the smaller set of elements, and the Conclusion is the larger set of elements.
If hypothesis then conclusion
hypothesis
Conclusion
If Mr. Erlin’s student, then Tam Student.
Mr. E’s
student
Tam Student
Example: A tiger is a type of cat.
Which category is bigger? Cats or Tigers. Cats, right? So the Euler Diagram looks like:
Cats
tiger
And the Conditional statement is:
If Tiger, then Cat. (Or, if an animal is a tiger, then it is a cat)
RSN03 Given a set of conditional statements, put them in order using a Logic Chain, and then
express them all as a single resulting conditional statement.
Notes:
Here are three Conditional Statements.
Notice that the blue words “pair up” and “cancel” each other out. So notice that TRET is
the only word that is unpaired and is a Hypothesis. Similarly, SORP is the only word that
is unpaired and is a Conclusion.
We can order these Conditional Statements into a Logic Chain by starting with the lone
hypothesis and ending with the lone conclusion.
Now, since we see the flow of Logic in the Chain, we can express the entire concept as a
single, summative Conditional Statement:
Think of it in terms of flights in an airplane.
if SFO then LAX
if LAX then BND
if BND then DUL
if DUL then MIA
So, we started in SFO and ended up in MIA…thus if SFO, then MIA
RSN 04 Given a conditional statement, write the corresponding Converse, Inverse and
Contrapositive statements.
Once we establish the conditional statement, and break it down to see the Hypothesis and the
Conclusion, we can re-order it into a couple of different forms. These forms will be important to
our ability to determine validity of a statement, but for now we are simply looking at creating the
forms themselves.
RSN05 Use converse, inverse, contrapositive to determine validity of conclusions made in
association with conditional statements.
When we create all four forms of the conditional statements, with various examples and contextual
scenarios, and assess their truth values (ie. Are the statements TRUE?), we discover that one of
two patterns ALWAYS appears:
1) Conditional is TRUE
2) Conditional is TRUE
Converse is FALSE
Converse is TRUE
Inverse is FALSE
Inverse is TRUE
Contrapositive is TRUE
Contrapositive is TRUE
We described this as the picture of the hamburger, where the buns are both always the same (and
true), and the two patties are always the same, (but may be either true or false). Given this, if we
KNOW that the conditional is TRUE, we can be sure that the contrapositive is also TRUE, but we
cannot be certain about the converse or inverse.
So, if I’m given an argument with a conclusion made, I can construct it in this
Conditional/Converse/Inverse/Contrapositive format, and then make decisions about whether or
not I can be certain about the conclusions.
For example:
1) If glorp, then swizzle. Glorp. Since the conditional is given (assumed to be true), and the
hypothesis (glorp) is further given as true, I know that I have a p=> ? scenario. I know
that p=>q is the conditional, and if the conditional is true, then the conditional is true. So I
can make a valid conclusion of “swizzle”.
2) If glorp then swizzle. Swizzle. Conditional is assumed to be true, this time the conclusion
(swizzle) is also true. I can start a conditional with conclusion known to be true (q=>?). I
know that q=>p is the converse, and that the converse isn’t reliably true all the
time…sometimes it is true, sometimes not. So from this info, I can make no conclusion.
3) If glorp then swizzle. Not glorp. Conditional is true, now “not p” is true. (~p=>?). I know
that ~p=>~q (inverse) isn’t reliably true just like converse. So no conclusion.
4) If glorp then swizzle. Not swizzle. Conditional is true, now “not q” is true. (~q=>?). I
know that ~q=>~p (contrapositive) IS true whenever the conditional is true. So I can
reliably know that ~p is also true…thus my conclusion is “Not glorp”. (One can also
reason that, since every time there is glorp, there is swizzle, and that in this case there is
no swizzle, then there can’t have been glorp…because if there had been glorp, then there
would have been swizzle.
RSN
06
Given a conditional statement, write a biconditional and determine its validity.
To write a bicondition, remove IF and THEN from the conditional, and stick IF AND
ONLY IF between the P(hypothesis) and Q (conclusion).
Ex: Conditional: If 90 degrees, then right angle.
Biconditional: 90 degrees if and only if right angle.
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08
A biconditional is a special kind of conditional statement. For biconditional to be valid,
both for the original conditional and all the other cases (converse, inverse and
contrapositive) must be True. Since we know that conditional and contrapositive always
have the same “truth value”, like converse and inverse always have the same “truth
value”, we really only need to verify that the conditional and the converse are both true.
Ex1: Is 90 degrees if and only if right angle a valid biconditional?
To check: the Conditional: if 90 degrees then right angle (true)
the Covnerse: if right angle then 90 degrees (true)
Ex2: Is Name is Stanley if and only if male a valid biconditional?
To check: the Conditional: if Name is Stanley then male (true)
the Covnerse: if male then Name is Stanley (False! Counter example
might be John is a male, but his name is not Stanley)
Determine whether a definition is good or not.
A definition is “good” if and only if it can be written as a Valid Biconditional.
Given a statement, determine the opposite or logical negation.
In geometry, we are concerned with all possibilities. So if we ask for the opposite of x=5,
the logical negation is x≠5, which means it could -5 or 1/5th, or 3/2 or 12,335. So the
logical negation of my pet is a dog is: my pet is NOT a dog (which could mean cat,
parrot, elephant, rock…any pet other than a dog).
When we use the logical negation on a conditional (p-> q), the result is (p -> ~q). Or, if I
want the opposite of “If I win the lottery then I’ll be rich”, it would be “If I win the
lottery then I wont be rich” (not just I’ll be poor).
RSN
09
Use Euler Diagrams to demonstrate whether a statement is a contradiction or not.
Ex: If Tam has an Open Campus, then we go to Grilly’s for lunch.
a) If Tam has Open Campus(
), can we make a conclusion?
Open Campus
!
Grilly’s
Yes, we can make a conclusion, because the star is also ALWAYS in Grilly’s, so
we DO go to Grilly’s for lunch.
b) If we do go to Grilly’s (
), can we make a conclusion?
?
Open Campus
Grilly’s
No, we can NOT make a conclusion, because the star may or may not be in the
Open Campus region.
c) If we do NOT have an Open Campus (
Open Campus
), can we make a conclusion?
?
Grilly’s
No, we can NOT make a conclusion, because while the star is definitely outside of
Open Campus, the star may or may not be in the Grilly’s region.
d) If we do NOT go to Grilly’s (
Open Campus
), can we make a conclusion?
Grilly’s
!
Yes, we can make a conclusion, because since the star is outside of Grilly’s it
MUST be outside of Open Campus…so we do NOT have Open Campus.
RSN
10
Given a set of conditions and related statements, apply indirect reasoning to come to a
conclusion. (aka Indirect Proof)
Three steps for indirect reasoning:
1) Assume the opposite (logical negation…p=>~q)
2) ~q likely has several scenarios, go about demonstrating why each and every one
of those is impossible or leads to a contradiction.
3) Since ~q is impossible, q MUST be true. So say what must be true.
Ex:
I wish to prove:
If I wish to increase my savings account more than just interest payments, then I must
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earn more than I spend.
1) Assume the opposite:
If I wish to increase my savings account more than just interest payments, then I
must NOT earn more than I spend.
2) Not earning more than I spend means earning exactly what I spend OR earning
less than I spend. If I earn exactly what I spend, then I have no money left to put
in savings, therefore this contradicts increasing my savings account. So I can’t
earn exactly what I spend. If I earn less than I spend, then I will put myself in
debt, and when I pay off my debt, I will reduce my savings account, again in
contradiction to my desire to increase savings account.
3) Therefore, if NOT earning more than I spend is impossible, it must be true that in
order to increase my savings account beyond interest payments I MUST earn
more than I spend.
Correctly match Properties of Equality and Congruence with representative examples.
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Given an algebraic equation, solve using a two column proof format.
RSN
13
Fill in the blanks to complete a two column proof of Overlapping Angles Theorem,
Overlapping Segments Theorems.