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Law of Syllogism

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Law of Syllogism

Study Guide

Key Definition

The Law of Syllogism states: If $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ is also true.

Important Notes

The Law of Syllogism is a form of deductive reasoning.
It allows the deduction of a direct relationship from two conditional statements.
Useful for simplifying complex logical arguments.
Make sure each statement logically follows from the previous.
It is essential to verify the truth of initial premises.

Mathematical Notation

$\rightarrow$ means 'implies'; the left side is the antecedent and the right side is the consequent

$∧$ means 'and' (logical conjunction)

$∀$ means 'for all' (universal quantifier)

Remember to use proper notation when solving problems

Why It Works

The law allows us to connect two related conditional statements and draw a new conclusion, simplifying logical reasoning with $(p \rightarrow q)$ and $(q \rightarrow r)$, yielding $(p \rightarrow r)$.

Remember

When using the Law of Syllogism, ensure that the middle term is the same in both premises, $q$.

Quick Reference

Law of Syllogism:$(p \rightarrow q) \land (q \rightarrow r) \rightarrow (p \rightarrow r)$

Understanding Law of Syllogism

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation: If $p \rightarrow q$ and $q \rightarrow r$, then $p \rightarrow r$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $p \rightarrow q$ and $q \rightarrow r$ are true, what does the Law of Syllogism conclude?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If you study hard, then you get good grades. If you get good grades, then you get a reward. What can you conclude?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider: If it rains, then the ground gets wet. If the ground gets wet, then the plants grow. What can be concluded?

Challenge Quiz

Single Choice Quiz

Advanced

Given $a \rightarrow b$, $b \rightarrow c$, and $c \rightarrow d$, what can you conclude?

Recap

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Review key concepts and takeaways