unit 6 because idk shit Flashcards
(35 cards)
Properties of inequalities 1
If a > b, and c ≥ d, then a + c > b + d (LIKE ADD. PROP.)
Properties of inequalities 1 EX
a = 5
b = 2
c = 3
d = 1
5 >2
+3 +1
8 > 3
Stays true
Properties of inequalities 2
If a > b and C > 0, then ac > bc and a/c > b/c
Properties of inequalities 2 EX
2
OR FOR NEGATIVES,
-2
Remember, sign FLIPS when dividing or multiplying negative numbers
Properties of inequalities 3
If a > b and c < 0, then ac < bc and a/c < b/c
Properties of inequalities 4
If a > b and b > c, then a > c (LIKE TRANS. PROP.)
Properties of Inequalities 5
If a = b + c and c > 0, then a > b (SUM IS GREATER THAN ITS PARTS”
PRACTICE: If XY = YZ + 5, then XY > YZ (T or F)
True
PRACTICE: If m<A = m<B +m<C, then m<b> m<C (T or F)</b>
False
PRACTICE: If m<H = m<J + m<K, then m<K> m<H</K>
False
PRACTICE: If 10 = y + 2, then y > 10
False
The Exterior Angle Inequality Theorem
The measure of exterior angles of a triangle is greater than the measures of either remote interior angle.
-Statement: If p, then q.
-What is the Inverse?
If not p, then not q
-Statement: If p, then q.
-What is the Contrapositive?
If not q, then not p
If the statement is true (or false),
Then the contrapositive is true (or false)
Statements and contrapositives of those statements are called what?
Logically equivalent statements
What does a conditional venn diagram look like?
Circle p is inside circle q
PRACTICE: (If p, then q.) All runners are athletes. Leroy is a runner. What’s the conclusion?
Leroy is an athlete.
PRACTICE: (If p, then q.) All runners are athletes. Lucia is not an athlete. What’s the conclusion?
Lucia is not a runner.
PRACTICE: (If p, then q.) All runners are athletes. Linda is an athlete. What’s the conclusion
No conclusion.
PRACTICE: (If p, then q.) All runners are athletes. Larry is not a runner. What’s the conclusion?
No conclusion.
What do indirect proofs start with?
“Assume that (opp. of what your proving is true)”
How to write an Indirect proof?
- Assume temporarily that the conclusion is not true
- Reason logically until you reach a contradiction of a known fact
- Point out that the temporary assumption is false and the conclusion is true.
PRACTICE:
Given: n is an integer and n^2 is even
Prove: n is even
SOMETHING LIKE THIS:
-Assume temporarily that n is not even
-Then n is odd and
n^2 = n x n
=odd x odd
=odd
This however contradicts the original given information that n^2 is even, meaning the assumption was false and that n is even
