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1.2 – Deduction: Making a Case
- Use deductive methods to test whether a rule or pattern is correct.
- Identify conditional statements and use them to form logical statements.
- Deduction – a way of thinking that starts with a given set of rules and conditions and figures out what must be true based on what is given.
- Known as deductive reasoning.
- Conditional Statement – statement that has the form “If A, then B,” where A is what you assume is true and B is the conclusion.
- Consecutive – the next one in the pattern.
- The n in nth stands for any counting number (positive numbers).
- The nth number in a series…
- To find the nth term:
- Divide n by however many numbers are in the pattern.
- Identify the remainder, R.
- If there is a blue marble, it is followed by 2 yellow marbles and a red marble.
- If there is a red marble, it is followed by a blue marble.
- What color is the 98th marble?
- Use the nth number pattern to solve this problem.
- We know there are 4 marbles in the pattern: blue, yellow, yellow, red.
- Divide 98 by 4, and find the remainder (R): 2
- The 2nd marble in the pattern is yellow, so the answer is yellow!
|Logic: Deduction vs Induction
- You CANNOT rely solely upon induction to prove that your conclusion is correct.
- You CAN rely solely upon deduction to prove that your conclusion is correct.
- In geometry, you can use deductive rules to PROVE conjectures.
- If A, then B
- Means: If A is true, then B is always true.
- Math Example: If 2 + 2 = 4, then 4 – 2 = 2.
- Life Example: If I study, then I will learn.
- Syllogism – a chain of conditional statements.
- How to Use a Chain of Conditional Statements
- Step 1: Start with a chain of conditional statements.
- Step 2: Accept that all of those statements are true.
- Step 3: Make connections between statements in the chain.
- Step 4: Make a conclusion statement based on those connections.
- All squares are rectangles.
- All rectangles are shapes that have four sides
- All shapes with four sides are quadrilaterals.
- So, all squares are quadrilaterals!
- Ex 1. If the pattern below follows the rule “Starting with five, every consecutive line has a number that is four less than twice the previous line,” how many marbles must be in the seventh line?
- Note: Notice that there are only 4 lines of marbles, but you need to find the number in the 7th line.
- Starting with 5 is not important in this case, as we already have the first few lines.
- Four less means: minus 4 (from something)
- Twice the previous line means: 2L
- Pattern: 2L – 4
- 5th Line: 2(12) – 4 = 24 – 4 = 20
- 6th Line: 2(20) – 4 = 40 – 4 = 36
- 7th Line: 2(36) – 4 = 72 – 4 = 68
- Ex 2. If the statement “If it is midnight, then the I am not awake” is assumed to be true, is its reverse, “If I am not awake, then it is midnight,” also always true?
- No, it is false! I might be napping around 2pm.
- Ex 3. The statement “If A, then B” can best be described as: “If A is true, then B MUST be true!”
- Ex 4. In deductive thinking, you start with a given set of rules and conditions and determine what must be true as a consequence.
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