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RSN 01Given a conditional (If, Then) statement, identify the hypothesis and the conclusion.
The conditional of a statement is the phrase following If (but not including it). The conclusion is the phrase following Then (but not including it). For example:
If you are driving in a school zone, then you are going 25 mph. Hypothesis: You are driving in a school zone. Conclusion: You are going 25 mph. |
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RSN 02Rewrite a statement in appropriate If Then conditional form.
You can easily convert a statement expressing a conditional relationship to a conditional statement using an Euler diagram like the one on the left. The inner circle represents the hypothesis of a statement. It is always contained within the outer circle, representing the conclusion of a statement. This shows visually that if a hypothesis is true, then a conclusion is automatically true, or else the statement is invalid.
Conditional relationship: The ground is wet whenever it rains. W: It rains P: The ground is wet If it rains, then the ground is wet. |
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Video by SmartBytz explaining conditional, converse, inverse, and contrapositive
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RSN 04Given a conditional statement, write the corresponding Converse, Inverse and Contrapositive statements.
To find the converse, inverse, and contrapositive of a conditional statement, you switch and/or negate (~) the hypothesis (p) and conclusion (q) of a statement in various ways:
conditional: p -> q If you play piano, then you have long fingers. converse: q -> p If you have long fingers, then you play piano. inverse: ~p -> ~q If you don't play piano, then you don't have long fingers. contrapositive: ~q -> ~p If you don't have long fingers, then you don't play piano. |
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RSN 05Use converse, inverse, contrapositive to determine validity of conclusions made in association with conditional statements.
There is a limited number of results you can get when assigning a truth value to various forms of a conditional. Find the converse, inverse, and contrapositive to exemplify these various results. For example:
conditional: If you are a dog, then you are a furry animal. converse: If you are a furry animal, then you are a dog. inverse: If you are not a dog, then you are not a furry animal. contrapositive: If you are not a furry animal, then you are not a dog. Here, we can assign truth values to these statements by using the always, sometimes, never technique of finding examples and counterexamples to prove and disprove various statements. Seeing as we can find counterexamples for the middle two (for instance, a cat), then the truth values are: Conditional: T Converse: F Inverse: F Contrapositive: T After analyzing other statements and truth values, we can see that statements are like the hamburger to the left: the conditional and the contrapositive (the buns) always match, as do the converse and the inverse (burgers, perhaps). However, it is only the buns (cond. and contra.) that are reliable when making conclusions. For example: If I run 100 miles per hour, I can break the world record. I did not run 100 miles per hour. What conclusion can you make, if any? I run 100 miles per hour is the hypothesis (p), which is being negated (~p). This means that it is the inverse of the conditional. However, because the inverse is not automatically true just because we've been told the conditional is true (a bun and a burger), we can make no conclusion. |
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RsN 06Given a conditional statement, write a biconditional and determine its validity.
A biconditional is a statement that can essentially be flipped around any which way and will still work.
For example (written in three different ways): It is a triangle if and only if it is a three-sided polygon. It is a triangle iff it is a three-sided polygon. (not a typo- iff means if and only if) Triangle (p) <-> three-sided polygon (q). Put in terms of conditional statements, all forms of the conditional have to be true for the biconditional to be true. However, because forms of the conditional are like a hamburger (RSN 05), you only have to test if the conditional and converse are true. Conditional: If I am in a country that uses the customary system of measurement, then I am in the USA, Myanmar, or Liberia. Converse: If I am in the USA, Myanmar, or Liberia, then I am in a country that uses the customary system of measurement. Since both have all examples and no counterexamples, both statements are said to be true. Therefore, the biconditional is true: I am in a country that uses the customary system of measurement iff I am in the USA, Myanmar, or Liberia. |
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Rsn 07Determine whether a definition is good or not.
A good definition is one where, if made into a conditional statement, then the biconditional is true (think of it as being as selective as possible). For example:
A triangle is a three-sided polygon. Biconditional: It is a triangle iff it is a three-sided polygon. Can you find counterexamples? If not, then it's a good definition. |
Rsn 08Given a statement, determine the “opposite ” (logical negation)
While many people would think that the negation of something is the inverse
(~p->~q), it is actually just the negation of the hypothesis (~p->q). What is the opposite? Black- not black (the answer "white" is wrong in logic, as incorrectly demonstrated by a chessboard) Up- not up (the answer down is incorrect) x>1- x≤1 (everything that the first expression doesn't include) |
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rsn 09Use Euler Diagrams to demonstrate whether a statement is a contradiction or not.
An Euler Diagram can show that a statement is a contradiction if an example can be both inside and outside a circle on the diagram. For instance, referencing the Euler diagram to the left:
If I go to Starbucks, then it is Sunday. If there can be an example that can prove the conditional and one that can disprove the conditional, then that form of the statement is a contradiction. They are when I go to Starbucks (example inside P) but it isn't Sunday (not inside W) or when I go to Starbucks (inside P) and it is Sunday (inside W). Since its examples can be both inside and outside, it is a contradiction. |
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RSN 10Given a set of conditions and related statements, apply indirect reasoning to come to a conclusion. (aka Indirect Proof)
Taking a conditional statement (p->q), you can prove that the statement is true using the following steps of Indirect Reasoning (a.k.a. Indirect Proof):
1. Assume the opposite, or logical negation (p->~q) 2. Reason to logical fallacy (e.g. pΛ~p) 3. Because the opposite is false, the statement must be true (p->q) Using an example: If there is a obtuse angle in a triangle, then there is only one. 1. p->~q: There are two or three obtuse angles in a triangle. 2. Reasoning to contradiction: If triangle, then angles add up to 180 degrees (Theorem proved later) 91+91+x=180 x=-2 Angles must be positive. 3. Because the logical negation has been proven to lead to logical fallacy, there can only be one obtuse angle in a triangle. |
If you are still struggling with the basic ideas of proofs, here's a good video by ukmathsteacher
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Rsn 12Given an algebraic equation, solve using a two column proof format.
This is just using simple algebra skills and justifying them with properties of equality. Essentially this skill is finally laying the foundation for writing your own proofs. For example: If 3x+8=14, then x=2
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Rsn 13Fill in the blanks to complete a two column proof of Overlapping Angles Theorem, Overlapping Segments Theorems.
This proof was already covered in ATM. Now that we have become more comfortable with proofs, you can take another look at this proof and then see if you can write the proof of the inverted version (If m∠AXB=m∠CXD, then m∠AXC=m∠BXD).
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