2. Nobody is despised who can manage a crocodile.
Babies are illogical.
Nobody is despised who can manage a crocodile.
Illogical persons are despised.
Lewis Carroll
Rewriting these in an If -then format helps to clarify the preceding statements.
If a person is a baby, then the person is not logical.
If a person is not despised, then that person can manage a crocodile.
If a person is not logical, then the person is despised.
What is the logical conclusion of these statements?
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3. If-then statements are called conditional statements or conditionals.
Hypothesis - is the If part (less the if), and the
Conclusion - is the then part (less the then) of the conditional.
If p, then q, where p and q are some statement, is represented symbolically with p  q.
Symbolically, if p, then q (original), becomes p  q
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4. Notice that all statements may not be an if-then statement format.
Parallel lines don’t intersect.
How would you arrange this to make an if-then statement?
If lines are parallel, then they do not intersect.
Linear pairs are supplementary.
How would you arrange this to make an if-then statement?
If two angles are a linear pair, then they are supplementary angles.
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5. I will go to your house, if it rains tomorrow.
When you are converting sentences to if-then statement, look for the key words, if and then. In our examples the if-part is always first, but some sentences may have the if-part at the end of the sentence. For example,
I will go to your house, if it rains tomorrow.
The hypothesis of this sentence is “it rains tomorrow” and the conclusion is “I will go to your house.” To make this a conditional statement it would be rearranged as follows.
If it rains tomorrow, then I will go to your house.
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6. If a person is a baby, then that person is illogical.
If a sentence has no if-then key words, then use the subject of the sentence as the hypothesis and the object of the sentence as the conclusion. An example is;
Babies are illogical.
becomes;
If a person is a baby, then that person is illogical.
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7. When p  q, then the converse is q  p.
The converse of a statement is when you exchange the hypothesis and the conclusion in a statement.
When p  q, then the converse is q  p.
If two lines are perpendicular, then they intersect.
and the converse is
If two lines intersect, then they are perpendicular.
Symbolically, p  q (original), becomes q  p
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8. Negation - The denial of a statement.
The angle is obtuse.
The denial of this statement is ;
The angle is not obtuse.
Other examples of mathematical statements and then their denial
Symbolically, p (original), becomes  p
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9. Vertical angles are congruent.
The inverse of a statement is formed by negating the hypothesis and the conclusion.
Vertical angles are congruent.
If two angles are vertical, then they are congruent.
and its inverse is,
If two angles are NOT vertical, then they are NOT congruent.
Symbolically, p  q (original), becomes  p   q
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10. If two lines are perpendicular, then they intersect.
The contrapositive of a statement is formed by negating the hypothesis and the conclusion of the converse.
original
If two lines are perpendicular, then they intersect.
converse
If two lines intersect, then they are perpendicular.
contrapositive
If two lines do not intersect, then they are not perpendicular.
Symbolically, p  q (original), becomes  q  p
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11. A logically equivalent statement has the form of p  q   q   p or
It can be proven that the contrapositive is logically equivalent to the original statement.
A logically equivalent statement has the form of
p  q   q   p or
if x2 > 4, then x > 2
if (x > 2), then (x2 > 4)
if x  2, then x2  4
if (3 = n + 1), then n = 2
if (n=2), then (3 = n + 1)
if n  2, then 3  n + 1
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12. Postulate 2-1, Through any two points there is exactly one line.
Postulate 2-2, Through any three points not on the same line there is exactly one plane.
Postulate 2-3, A line contains at least two points.
Postulate 2-4, A plane contains at least three points not on the same line.
Postulate 2-5, If two points lie in a plane, then the entire line containing those two points lies in that plane.
Postulate 2-6, If two planes intersect, then their intersection is a line.
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13. If you are 13 years old, then you are a teenager.
converse **
If you are a teenager, then you are 13 years old.
inverse **
If you are NOT 13 years old, then you are NOT a teenager.
contrapositive
If you are NOT a teenager, then you are NOT 13 years old.
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14. If an angle measures 90o, then it is a right angle.
converse
If an angle is a right angle, then it measures 90o.
inverse
If an angle is NOT a right angle, then it does NOT measure 90o.
contrapositive
If an angle does NOT measure 90o, then is NOT a right angle.
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15. Summary
We converted written statements into conditionals (if-then statements) in order to use logic to determine the validity (truth or falsehood) of those statements.
The form of the if-then is if p, then q, or p  q, where the if-part of the conditional is the hypothesis and the then-part is the conclusion.
We discussed the converse, the negation, the inverse, and the contrapositive of a conditional. We found that the inverse and the converse of a true conditional, is not always itself true. We also discovered that the conditional and its contrapositive are logically equivalent.
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16. END OF LINE
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