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Deductive reasoning is the type of reasoning used when making a Geometric proof, when attorneys present a case, or any time you try and convince someone using facts and arguments. How to define deductive reasoning and compare it to inductive reasoning? Example: 1. Prove QUAD is a parallelogram. 2. Draw the next shape. Show Step-by-step Solutions
- Mathematical
- Recognizing Flawed Reasoning
- SOLUTION
- MAKING MATHEMATICAL ARGUMENTS
- Conditional Statement
- Rewriting a Statement in If-Then Form
- SOLUTION
- Negation
- Writing a Negation
- SOLUTION
- Converse
- COMMON ERROR
- Inverse
- Contrapositive
- Symbols
- Writing Related Conditional Statements
- SOLUTION S
- Using Defi nitions
- perpendicular lines.
- Using Defi nitions
- Biconditional Statement
Mathematically profi cient students communicate mathematical ideas,
The syllogisms below represent common types of fl awed reasoning. Explain why each conclusion is not valid. a. When it rains, the ground gets wet. b. If ABC is equilateral, then it is isosceles. △ The ground is wet. ABC is not equilateral. △ Therefore, it must have rained. Therefore, it must not be isosceles. c. All squares are polygons. d. No tri...
The ground may be wet for another reason. A triangle can be isosceles but not equilateral. All squares are quadrilaterals, but not because all trapezoids are quadrilaterals. No squares are triangles.
To be profi cient in math, you need to distinguish correct logic or reasoning from that which is fl awed.
A conditional statement is a logical statement that has two parts, a hypothesis p and a conclusion q. When a conditional statement is written in if-then form, the “if” part contains the hypothesis and the “then” part contains the conclusion.
Use red to identify the hypothesis and blue to identify the conclusion. Then rewrite the conditional statement in if-then form. a. All birds have feathers. b. You are in Texas if you are in Houston.
All birds have feathers. You are in Texas if you are in Houston.
The negation of a statement is the opposite of the original statement. To write the negation of a statement p, you write the symbol for negation ( ∼) before the letter. So, “not p” is written ∼p.
Write the negation of each statement. The ball is red. The cat is not black.
The ball is not red. The cat is black. Core Concept
To write the converse of a conditional statement, exchange the hypothesis and the conclusion.
Just because a conditional statement and its contrapositive are both true does not mean that its converse and inverse are both false. The converse and inverse could also both be true.
To write the inverse of a conditional statement, negate both the hypothesis and the conclusion.
To write the contrapositive of a conditional statement, fi rst write the converse. Then negate both the hypothesis and the conclusion.
∼q → ∼p A conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. In general, when two statements are both true or both false, they are called equivalent statements.
Let L p be “you are a guitar player” and let q be “you are a musician.” Write each statement s in words. Then decide whether it is true or false. a the conditional statement p q → b the converse q p → c the inverse p q ∼ → ∼ d the contrapositive q p ∼ → ∼
a Conditional: If you are a guitar player, then you are a musician. true; Guitar players are musicians. Converse: If you are a musician, then you are a guitar player. false; Not all musicians play the guitar. Inverse: If you are not a guitar player, then you are not a musician. false; Even if you do not play a guitar, you can still be a musician. C...
You can write a defi nition as a conditional statement in if-then form or as its converse. Both the conditional statement and its converse are true for defi nitions. For example, consider the defi nition of perpendicular lines. If two lines intersect to form a right angle, then they are
You can also write the defi nition using the converse: If two lines are perpendicular lines, then they intersect to form a right angle. You can write “linel is perpendicular to line m” asl m. ⊥ m m ⊥
Decide whether each statement about the diagram is true. Explain your answer using the defi nitions you have learned. a. AC BD ⃖ ⃗ ⊥ ⃖ ⃗ b. ∠AEB and ∠CEB are a linear pair.
When a conditional statement and its converse are both true, you can write them as a single biconditional statement. A biconditional statement is a statement that contains the phrase “if and only if.” Words p if and only if q Symbols p q ↔ Any defi nition can be written as a biconditional statement. Writing a Biconditional Statement Rewrite the def...
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MEASUREMENT AND GEOMETRY • GEOMETRIC REASONING 6 a Complete this statement: AB AD = BC AE. b Find the value of the pronumerals. 7 a Find the values of h and i. b Find the values of j and k. 8 Find the value of the pronumeral in the diagram at right. 9 The triangles shown at right are similar. Find the value of x and y.
Definition of Isosceles Trapezoid: A trapezoid in which the base angles and non-parallel sides are congruent Statements overlapping triangles Reasons 1. Given 2. Definition of midpoint 3. Definition of midpoint 4. Division property ("like division" of congruent segments) 5. Reflexive property 6. Side-Angie-Side (SAS) (4, 5, 1) 7. CPCTC Reasons ...
Deductive reasoning is the process of reasoning from accepted facts to a conclusion. In contrast to deductive reasoning is inductive reasoning (thinking). Consider a golfer who tries a new grip on his clubs. The next 4 times he plays golf, he strokes the ball better, he decides its because of the new grip. That’s an example of inductive ...
Emerging deductive Reasoning - Work analytically with properties of rectangles. Beginning to recognise necessary and sufficient conditions. Use sound reasoning in argument/explanations, though explanations often are procedurally based or base on an example. Able to recognise the relationship between length, area and volume.
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Chapter 2 Notes: Reasoning and Proof Page 1 of 3 2.1 – Inductive Reasoning . Defined Terms . Conjecture – An unproven statement that is based on observations. Inductive Reasoning – Finding a pattern in specific cases and then making a conjecture for the general case. Counterexample – A specific case for which a conjecture is false ...
