TYPES OF "IF, THEN" STATEMENTS:
CONDITIONAL• if p, then q
-usually true -reliable -always same as contrapositive (true or false) |
CONVERSE• if q, then p
-always same as inverse (true or false) -unreliable |
INVERSE• if not p, then not q (not can be a single squiggly line)
-always same as converse -unreliable |
CONTRAPOSITIVE• if not q, then not p
-always same as conditional -reliable |
BICONDTIONAL:
![Picture](/uploads/5/9/2/3/59231587/8374485.jpeg?1444708023)
A biconditional is true if the conditional and converse are both true. A biconditional if written as: p if and only if (or iff) q
This means it goes both ways: if p then q , and if q then p
EX: If an angle has exactly 90 degrees, then it's a right angle
-conditional: true
-converse: if an angle is a right angle, then it has exactly 90 degrees: true
-both are true, so...
-BICONDITIONAL: an angle is right iff it has exactly 90 degrees
Here's a great video that explains this:
http://www.virtualnerd.com/geometry/reasoning-proof/conditional-biconditional-statements/biconditional-statement-definition
This means it goes both ways: if p then q , and if q then p
EX: If an angle has exactly 90 degrees, then it's a right angle
-conditional: true
-converse: if an angle is a right angle, then it has exactly 90 degrees: true
-both are true, so...
-BICONDITIONAL: an angle is right iff it has exactly 90 degrees
Here's a great video that explains this:
http://www.virtualnerd.com/geometry/reasoning-proof/conditional-biconditional-statements/biconditional-statement-definition
GOOD DEFINITION:
A biconditional that is true is a good definition. So, in order to determine if a definition is good or not, put in in a conditional form. If the conditional is true, test out all the other statements. If all 4 are true, then the biconditional has to be true, which means it's a good definition. You can also just look at the conditional and converse, because the inverse and contrapositive will be true or false, depending on these. If the conditional and converse are true, then it's a good definition. (Want to be super confused? This purple definition is a good definition, too!)
LOGIC CHAIN:
![Picture](/uploads/5/9/2/3/59231587/769807988.jpg?310)
We use logic chains to organize a set of If, Then statements and we can use this to write an overall if then statements.
Here's an example of a set of if then statements:
If the sprinkler goes, then the grass gets watered.
If the grass gets watered, then the grass is green.
If I turn on the sprinkler, then the sprinkler starts.
If the grass is green, then our yard looks nice.
I can use a logic chain to organize them. I do this by matching up the conclusions with the hypothesis, for example:
"...then the grass gets watered" matches up with "If the grass gets watered", so I keep connecting pieces until I find the first statement. Next to this statement, I would write a 1 to show that it goes first in the logic chain.
The final order would look like this: Next, we can rearrange these statements in the correct order:
If the sprinkler starts, then the grass gets watered. 2 If I turn on the sprinkler, then the sprinkler starts.
If the grass gets watered, then the grass is green. 3 If the sprinkler starts, then the grass gets watered.
If I turn on the sprinkler, then the sprinkler starts. 1 If the grass gets watered, then the grass is green.
If the grass is green, then our yard looks nice. 4 If the grass is green, then our yard looks nice.
Now, we can use the first hypothesis and the last conclusion to make an overall if then statement:
If I turn on the sprinkler, then our yard looks nice.
Here's an example of a set of if then statements:
If the sprinkler goes, then the grass gets watered.
If the grass gets watered, then the grass is green.
If I turn on the sprinkler, then the sprinkler starts.
If the grass is green, then our yard looks nice.
I can use a logic chain to organize them. I do this by matching up the conclusions with the hypothesis, for example:
"...then the grass gets watered" matches up with "If the grass gets watered", so I keep connecting pieces until I find the first statement. Next to this statement, I would write a 1 to show that it goes first in the logic chain.
The final order would look like this: Next, we can rearrange these statements in the correct order:
If the sprinkler starts, then the grass gets watered. 2 If I turn on the sprinkler, then the sprinkler starts.
If the grass gets watered, then the grass is green. 3 If the sprinkler starts, then the grass gets watered.
If I turn on the sprinkler, then the sprinkler starts. 1 If the grass gets watered, then the grass is green.
If the grass is green, then our yard looks nice. 4 If the grass is green, then our yard looks nice.
Now, we can use the first hypothesis and the last conclusion to make an overall if then statement:
If I turn on the sprinkler, then our yard looks nice.
OPPOSITE (logical negation):
![Picture](/uploads/5/9/2/3/59231587/269509119.gif?132)
When we need to use the opposite of a statement, like the statement "the ball is red", we don't just say that "the ball IS NOT red", we have to use the logical negation. The correct opposite of "the ball is red" is actually "the ball is any color other than red, because this includes every possible other scenario for the ball. "the ball is not red" is too vague, and we need to show that we are negating the whole sentence, and naming all the other possible colors the ball could be.
Here's another example:
If we have the statement, x is less than 5, the logical negation isn't "x isn't less than 5", it would actually be every other possible scenario except x is less than 5, and the only two possible outcomes besides this are: x is greater than 5 AND x equals 5, so you have to include both of these. To combine the equal and greater than signs, you do this: (picture to the right), so the opposite of x is less then 5 is: x is greater than or equal to 5
Here's another example:
If we have the statement, x is less than 5, the logical negation isn't "x isn't less than 5", it would actually be every other possible scenario except x is less than 5, and the only two possible outcomes besides this are: x is greater than 5 AND x equals 5, so you have to include both of these. To combine the equal and greater than signs, you do this: (picture to the right), so the opposite of x is less then 5 is: x is greater than or equal to 5
INDIRECT REASONING:
To prove that something is true, like "Jerry killed that man", we can follow a series of steps:
1. Assume the opposite
Ex: Jerry is NOT the killer
2. Take all the possibilities left and reason to contradiction
Ex: We know the killer was wearing a gray jacket, so it can't be these 3 people, we know the killer had a mustache, so it can't be that person, etc.
3. When there is only one option left, that must be the answer
Ex: Since Jerry would be the only person left over after all these clues, we know he MUST have killed that man.
1. Assume the opposite
Ex: Jerry is NOT the killer
2. Take all the possibilities left and reason to contradiction
Ex: We know the killer was wearing a gray jacket, so it can't be these 3 people, we know the killer had a mustache, so it can't be that person, etc.
3. When there is only one option left, that must be the answer
Ex: Since Jerry would be the only person left over after all these clues, we know he MUST have killed that man.
TRUTH TABLES:
We use truth tables to understand if a statement is true or false. You always start by filling in the hypothesis column (p) with true, true, false, false (T, T, F, F) and the conclusion column (q) with true, false, true, false (T, F, T, F). This will give you all the possible scenarios for your if, then statement. We use symbols, like the ones above, so it's easy to fill out a truth table.
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EULER (pronounced oiler) DIAGRAMS:
PROPERTIES OF EQUALITY AND CONGRUENCE:
The image to the left shows the properties of equality. You add "of equality" to the end, like: "Addition Property of Equality". There are also examples for each property. The properties of congruence are:
-Symmetric Property of Congruence -Reflexive Property of Congruence -Transitive Property of Congruence There are examples for each of these below the first image. |
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We can use these Properties for... PROOFS!
We can use two column proofs to show our understanding for things like equations, theorems, and postulates. A couple notes to understand about proofs:
-You should always have two columns: statements and reasons. You write the math or actual writing under statements, and the reason WHY you do this under reasons.
-You always start with the "Given" statement
-You never, ever, ever, ever use "prove" as a reason! You always have to explain it
-You write "Q.E.D" after you finish a proof, which is a shortened latin phrase that mean, "That which was to be proven." (You're basically showing the proof is done)
Here are some examples of proofs: Algebraic Equation, Segment Overlapping Theorem, and Angle Overlapping Theorem
-You should always have two columns: statements and reasons. You write the math or actual writing under statements, and the reason WHY you do this under reasons.
-You always start with the "Given" statement
-You never, ever, ever, ever use "prove" as a reason! You always have to explain it
-You write "Q.E.D" after you finish a proof, which is a shortened latin phrase that mean, "That which was to be proven." (You're basically showing the proof is done)
Here are some examples of proofs: Algebraic Equation, Segment Overlapping Theorem, and Angle Overlapping Theorem