# 1.2 – Deduction: Making a Case

## Objectives

• Use deductive methods to test whether a rule or pattern is correct.
• Identify conditional statements and use them to form logical statements.

## Key Terms

• 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.

## Notes

Patterns
• The n in nth stands for any counting number (positive numbers).
• The nth number in a series…
• To find the nth term:
1. Divide n by however many numbers are in the pattern.
2. Identify the remainder, R.

Example

Rules

1. If there is a blue marble, it is followed by 2 yellow marbles and a red marble.
2. If there is a red marble, it is followed by a blue marble.
3. 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.

Conditional Statements
• 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 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.

Example 1

Example 2

• All squares are rectangles.
• All rectangles are shapes that have four sides
• All shapes with four sides are quadrilaterals.
• So, all squares are quadrilaterals!

## Examples

 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   Answer 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.