Math elec: 1st

Conditional Statement

> can be true or false

counter example

> in which the hypothesis is fulfilled while the conclusion is not.

• Conjunction (and):

  • Represents the logical "and" operation.

  • A conjunction is true only if both parts are true.

  • Symbol: ∧

  • Example: "The sun is shining and the birds are singing."

• Disjunction (or):

  • Represents the logical "or" operation.

  • A disjunction is true if at least one part is true (or if both are true). This is often referred to as an "inclusive or".

  • Symbol: ∨

  • Example: "I will eat an apple or I will eat a banana."

• Conditional (if...then):

  • Represents the logical "if...then" operation.

  • A conditional is false only if the "if" part (antecedent) is true and the "then" part (consequent) is false.

  • Symbol: →

  • Example: "If it rains, then the ground will be wet."

• Simple (or atomic) statements:

  • These are statements that cannot be broken down further into simpler statements using logical connectives.

  • These are the basic building blocks of compound statements.

  • Example: "The cat is black." or "2+2=4".

  • Your usage of the word "is" is correct in that it is often the verb that connects the subject of a simple statement to it's predicate.

  CONVERSE,INVERSE, AND CONTRAPOSITIVE

• Conditional Statement (If p, then q):

  • This is the original statement.

  • "p" is the antecedent (hypothesis).

  • "q" is the consequent (conclusion).

• Converse (If q, then p):

  • The converse switches the antecedent and consequent.

  • It is not logically equivalent to the original conditional. Meaning, if the original is true the converse can be either true or false.

• Inverse (If ~p, then ~q):

  • The inverse negates both the antecedent and consequent of the original conditional.

  • The inverse is also not logically equivalent to the original conditional.

• Contrapositive (If ~q, then ~p):

  • The contrapositive negates both the antecedent and consequent and then switches them.

  • The contrapositive is logically equivalent to the original conditional. This means if the original is true, the contrapositive is always true, and vice versa.

Example:

• Conditional: If it is raining (p), then the ground is wet (q).

• Converse: If the ground is wet (q), then it is raining (p).

• Inverse: If it is not raining (~p), then the ground is not wet (~q).

• Contrapositive: If the ground is not wet (~q), then it is not raining (~p).

Converse: If a triangle is scalene, then it has no equal sides.

Inverse: If a triangle has equal sides, then it is not scalene.

Contrapositive: If a triangle is not scalene, then it has equal

sides.

biconditional statement

> If a conclusion statement is true and its converse is also true

Determine the converse, inverse, and contrapositive of an if-then statement:

Converse: If the difference of two numbers is one, then they are consecutive whole numbers.

Inverse: If two numbers are not consecutive whole numbers, then their difference is not one.

Contrapositive: If the difference between two numbers is not one, then they are not consecutive whole numbers.

REASONING

REASONING

> A process based on experience and principles that allow one to arrive at a conclusion.

Intuition:

• Definition:

  • - A sudden insight allows one to make a statement without applying any formal reasoning.

    - “Jumping” to conclusions

    - In a cartoon, the character having the “bright idea”(using intuition) is shown with a light bulb next to her or his head.

• Example:

  • You walk into your geometry class, look at the teacher, and conclude that you will have a quiz today.

2. Induction:

• Definition:

  • We often use specific observations and experiments to draw a general conclusion.

  • As you would expect, the observation/experimentation process is common in laboratory and clinical settings.

• Characteristics:

  • Mostly used in creating generalization which are called conjectures.

• Example:

  • "Every time I've eaten peanuts, I've had an allergic reaction. Therefore, I'm probably allergic to peanuts."

  • Every 100 ml carton of yogurt in the store costs 30 pesos.

3. Deduction:

• Definition:

  • Deductive reasoning involves drawing specific conclusions from general principles or premises.

  • Type of reasoning in which the knowledge and acceptance of selected assumptions guarantee the truth of a particular conclusion.

  • A kind of reasoning that starts from a general statement to a particular statement

    Characteristics:

  • It's based on logical rules.

  • Its conclusions are certain if the premises are true.

  • It's used to prove theorems and solve problems.

• Example:

  • "All humans are mortal. Socrates is a human. Therefore, Socrates is mortal."

  • If a student plays on the Diliman High School boys’ varsity basketball team, then he is a talented athlete.

    2. Ted plays on the Diliman High School boys’ varsity basketball team.

    Conclusion: Ted is a talented athlete.

ARGUMENTS

Valid Argument:

• Definition:

  • A valid argument is one in which, if the premises are true, then the conclusion must also be true.

  • Validity is about the structure of the argument, not the actual truth of the premises.

  • In a valid argument, it is impossible for the premises to be true and the conclusion to be false.

• Key Points:

  • Validity is a property of the argument's form.

  • A valid argument can have false premises and a false conclusion, but it cannot have true premises and a false conclusion.

  • Examples of valid argument forms include:

    • Modus ponens (If P, then Q. P. Therefore, Q.)

    • Modus tollens (If P, then Q. Not Q. Therefore, not P.)

• Example:

  • Premise 1: All cats are mammals.

  • Premise 2: Mittens is a cat.

  • Conclusion: Therefore, Mittens is a mammal.

  • This is a valid argument, because if the premises are true, the conclusion must also be true.

Invalid Argument:

• Definition:

  • An invalid argument is one in which the conclusion does not necessarily follow from the premises, even if the premises are true.

  • In an invalid argument, it is possible for the premises to be true and the conclusion to be false.

• Key Points:

  • Invalidity indicates a flaw in the argument's structure.

  • Examples of invalid argument forms include:

    • Affirming the consequent (If P, then Q. Q. Therefore, P.)

    • Denying the antecedent (If P, then Q. Not P. Therefore, not Q.)

• Example:

  • Premise 1: If it is raining, then the ground is wet.

  • Premise 2: The ground is wet.

  • Conclusion: Therefore, it is raining. ¹  

  • This is an invalid argument. The ground could be wet for reasons other than rain (e.g., a sprinkler).