VA SOL Standard G.RLT.1
Your parent says, "If you clean your room, then we can go to the Mill Mountain Star."
Example (I Do): "Roanoke is the capital of Virginia." This is a proposition. Its truth value is False.
Your Turn 1: "What is your name?"
Your Turn 2: "A square has four right angles."
Example (I Do): Let p: "A triangle has three sides." (True)
The negation, ~p, is "A triangle does not have three sides." (False)
Your Turn 1: Let p: "It is raining."
Your Turn 2: Let q: "All angles are acute." (False)
The negation of "all" is "not all" or "some are not."
Example (I Do): Let p: "The sky is blue" (True) and q: "The grass is purple" (False).
Your Turn 1: Let p: "A rectangle has four sides" (True) and q: "A rectangle has four right angles" (True).
Your Turn 2: Let p: "I will study" and q: "I will fail".
A conditional statement is a logical argument that links two parts: a hypothesis and a conclusion. It is often called an "if-then" statement or an "implication."
Example (I Do): Identifying Parts
Statement: "If a figure is a square, then it has four right angles."
Your Turn 1: Writing in If-Then Form
Rewrite the sentence "All birds have feathers" as a conditional statement.
Your Turn 2: Identifying Parts
Statement: "You can drive in Virginia if you have a driver's license."
A conditional statement is only considered false when a true hypothesis fails to produce the promised true conclusion. Think of it like a promise.
Example (I Do): Truth Table
Promise: "If you clean your room (p), then I will give you $10 (q)."
| p (You Clean) | q (I Pay) | Reasoning | |
|---|---|---|---|
| True | True | True | You cleaned, I paid. Promise kept. |
| True | False | False | You cleaned, I didn't pay. Promise BROKEN. |
| False | True | True | You didn't clean, but I paid anyway. No promise was broken. |
| False | False | True | You didn't clean, and I didn't pay. No promise was broken. |
Your Turn 1: Let p: "2+2=4" (True) and q: "Earth is flat" (False).
Your Turn 2: Let p: "Pigs can fly" (False) and q: "Roanoke is in Virginia" (True).
A true statement is: "If a pet is a dog, then it is a mammal." Is the statement "If a pet is a mammal, then it is a dog" also always true?
From any conditional statement (p → q), we can form three related statements by rearranging and negating the hypothesis and conclusion.
Example (I Do): Let's analyze all four forms for a single statement.
Original Conditional: If an angle is a right angle, then its measure is 90°. (True)
Your Turn 1: Given "If a shape is a square, then it is a rectangle."
Your Turn 2: Given "If a number is divisible by 4, then it is even."
Logically equivalent statements are statements that always have the same truth value.
Example (I Do): "An angle is a right angle if and only if its measure is 90°." This is a valid biconditional because both the conditional and its converse were true in the example above.
Your Turn 1: Consider the conditional: "If two lines are perpendicular, then they form congruent adjacent angles."
Your Turn 2: Consider the conditional: "If a polygon is a square, then it is a rectangle."
Write the converse and contrapositive of: "If two angles are complementary, then their sum is 90°."
What was the most important concept you learned today? What concept is still confusing, or what questions do you have?
You see muddy footprints leading away from the kitchen door. You also know that it rained earlier. What is a logical conclusion you can make?
Example (I Do): A Valid but Untrue Argument
"If you are a happy person, then you like animals. If you like animals, then you like dogs. Therefore, if you are a happy person, then you like dogs."
The logical structure is valid, but the premises themselves may not be true for everyone.
In mathematics, we use two main types of reasoning to make and prove claims.
Example (I Do):
Your Turn 1: A scientist tests a drug on 1,000 patients and it works for 990 of them. She concludes the drug is effective. This is reasoning.
Your Turn 2: The speed limit is 70 mph. You are driving 80 mph. You conclude you are speeding. This is reasoning.
Deductive reasoning follows specific structures, or "laws," to ensure the conclusion is valid. The symbol for "therefore" is ∴.
The Law of Syllogism is a form of deductive reasoning. It states that if two conditional statements are true, and the conclusion of the first is the same as the hypothesis of the second, you can create a new true conditional statement by linking the first hypothesis to the final conclusion.
Structure: If $p \rightarrow q$ and $q \rightarrow r$ are true, then you can conclude $p \rightarrow r$. ∴ p → r
Example (I Do):
Conclusion: ∴ "If it's a holiday (p), then I can sleep in (r)."
Your Turn 1:
Given: "If a figure is a square, then it is a rhombus."
And Given: "If a figure is a rhombus, then its diagonals are perpendicular."
Your Turn 2:
Given: "If I save my money, I can buy a car."
And Given: "If I buy a car, I can get a job."
The Law of Detachment is a form of deductive reasoning. It states that if a conditional statement is true and its hypothesis is true, then its conclusion must also be true.
Structure: If $p \rightarrow q$ and $p$ are true, then you can conclude $q$. ∴ q
Example (I Do):
Conclusion: ∴ "Maria passed Algebra 1."
Your Turn 1:
Given: "If a shape is a square, then it has four right angles."
And Given: "The shape is a square."
Your Turn 2:
Given: "If you live in Roanoke, then you live in Virginia."
And Given: "You live in Roanoke."
Example (I Do): Statement: "If a number is prime, then it is odd."
This is false. A counterexample is the number 2. The hypothesis "2 is prime" is true, but the conclusion "2 is odd" is false.
Your Turn 1: Disprove "If a shape has four sides, then it is a square."
Your Turn 2: Disprove "If a city is in Virginia, then its name is Roanoke."
Given: "If Maria goes to the mall, she will buy shoes." and "Maria went to the mall."
What was the most important concept you learned today? What concept is still confusing, or what questions do you have?
Think about your friends. Some play video games, and some play sports. Some do both. How would you draw a picture to show this relationship?
Venn diagrams provide a visual way to understand logical operators.
Intersection: $A \cap B$ ("AND")
Union: $A \cup B$ ("OR")
Complement: $A'$ ("NOT")
Example (I Do): A survey of 25 students finds 14 play soccer and 10 play basketball. If 6 play both, how many play neither?
Your Turn 1: In a class of 30 students, 18 take Art, 15 take Music, and 7 take both.
Your Turn 2: 50 people were surveyed. 30 liked summer and 25 liked winter. 12 liked both.
A survey of 100 students at a Roanoke high school found that 60 play an instrument and 45 sing in the choir. If 20 students do both, how many students do neither?
What was the most important concept you learned today? What concept is still confusing, or what questions do you have?
Test your knowledge. You have 3 attempts.
1. Let p: "Roanoke is in Virginia" (True) and q: "The Mill Mountain Star is blue" (False). What is the truth value of the conjunction $p \land q ? $
2. Let p: "It is raining" and q: "I will take an umbrella." Which symbolic statement represents "If it is raining, then I will NOT take an umbrella"?
3. What is the contrapositive of the statement "If a shape is a rectangle, then it has four sides"?
4. A statement can be written as a true biconditional if the original conditional is true and its ______ is also true.
5. Using definitions, postulates, and theorems to prove a conclusion is an example of what type of reasoning?
6. Given: "If two angles are vertical, then they are congruent." and "∠A and ∠B are vertical." What is the logical conclusion by the Law of Detachment?
7. Given: "If I save money, I can buy a car." and "If I buy a car, I will drive to work." What can you conclude by the Law of Syllogism?
8. A survey of 50 students found 20 play soccer and 28 play baseball. If 8 students play both sports, how many students play only soccer?
9. The set of all elements that are in Set A OR Set B is called the...
10. Which statement is logically equivalent to "If you live in Virginia, then you live in the USA"?