Geometry Unit 2- Logic

Logic Staments

Conditional Statements

  • contains a hypothesis (p) and a conclusion (q),

in the format: if p, then q

Converse Statements

  • switches the hypothesis and conclusion

  • in the format if q, then p

Inverse Statements

  • negates hypothesis and conclusion, resulting in the structure if not p, then not q.

Contrapositive Statements

  • negates and switches the hypothesis and conclusion, leading to the structure if not q, then not p.

Biconditional Statements

  • A biconditional statement is true and can only be written if both the conditional and the converse are true or structured as "p if and only if q."

Conjecture

  • based on observations and needs to be proven through logical reasoning and valid proof to determine its truth value.

Counterexample

  • to disapprove a conjecture


Logic Symbols

not = negation, often represented as ~ in logical expressions.

or = is usually a v or u in logic statements

and = is usually an upside down v or u in logic statements

therefore = is represented with 3 dots that form a triangle

if and only if = represented as ⇔\lrArr


Laws of Logic

  1. Law of Detachment

This law states that

Given: P →\rightarrow Q

P

Conclusion: Q

  1. Law of Syllogism

This law states that

Given: P →\rightarrow Q

Q →\rightarrow R

Conclusion: P →\rightarrow R

(it could also be expressed as p to q, r to p, then r to q)

  1. Law of Contrapositive

This law states that

Given: P →\rightarrow Q

Conclusion: ~Q →\rightarrow ~P

Venn Diagram

Venn Diagram is a way to visually represent logical relationships between different sets, allowing for a better understanding of the implications of the given statements.

The area where P and Q overlap indicates instances where both conditions are true, while the areas outside the circles highlight conditions that are not satisfied, reinforcing the concept of contraposition in logical reasoning. This visual aid helps clarify how the truth values of P and Q interact, providing insight into the validity of arguments and simplifying the process of understanding complex logical propositions.