True and False Statements

A **statement** is a sentence that is either **true** or **false**, but not both. In mathematics and logic, we work with statements that can be clearly evaluated: **True statements** are correct. They match reality or follow mathematical rules: - "$2 + 3 = 5$" is **true** - "A square has 4 sides" is **true** **False statements** are incorrect. They do not match reality: - "$2 + 3 = 7$" is **false** - "A triangle has 5 sides" is **false** Not every sentence is a statement! Questions ("What is $2 + 3$?") and commands ("Add these numbers") are NOT statements because they cannot be true or false.

Evaluating a Simple Math Statement

Is the statement "$8 + 5 = 13$" true or false?

Identify the statement: The statement claims that 8 plus 5 equals 13 = $8 + 5 = 13$

Calculate the left side: $8 + 5 = 13$ = 13

Compare with the right side: Left side: 13, Right side: 13 = 13 = 13

Determine if true or false: Both sides are equal = TRUE

Answer: The statement "$8 + 5 = 13$" is TRUE.

Evaluating a False Statement

Is the statement "$7 \times 4 = 26$" true or false?

Identify the claim: The statement claims that 7 times 4 equals 26 = $7 \times 4 = 26$

Calculate the left side: $7 \times 4 = 28$ = 28

Compare with the right side: Left side: 28, Right side: 26 = $28 \neq 26$

Determine if true or false: The sides are NOT equal = FALSE

Answer: The statement "$7 \times 4 = 26$" is FALSE. The correct value is 28.

Inequality Statements

Is the statement "$15 > 9$" true or false?

Read the statement: The statement claims 15 is greater than 9 = $15 > 9$

Compare the numbers: On the number line, bigger numbers lie further to the right = 15 is to the right of 9

Evaluate the claim: Yes, 15 is greater than 9 = TRUE

Answer: The statement "$15 > 9$" is TRUE.

Statements About Shapes

Is the statement "All rectangles have 4 equal sides" true or false?

Understand the claim: The statement says every rectangle has 4 sides that are all the same length = Claim about rectangles

Recall rectangle properties: A rectangle has 4 sides and 4 right angles. Opposite sides are equal. = Opposite sides equal

Find a counterexample: A rectangle can be 5 cm by 3 cm - opposite sides equal, but not all 4 sides equal = Not all sides equal

Conclusion: Only squares have 4 equal sides. Rectangles do not always. = FALSE

Answer: The statement is FALSE. Rectangles have opposite sides equal, but not all 4 sides equal (except for squares).

Understanding true and false statements is the foundation of logical thinking: - **Problem Solving**: When you check your math work, you evaluate whether your answer makes the statement true - **Science**: Scientists test hypotheses to determine if they are true or false - **Computer Programming**: Computers use true/false logic to make decisions - **Daily Life**: You constantly evaluate claims - "The store closes at 9 PM" is either true or false Learning to identify and evaluate statements helps you think critically and make better decisions!

Mistake: Thinking questions can be true or false

Why: "What is 5 + 3?" is a question, not a statement. It asks for information rather than making a claim.

Correct: Only declarative sentences that make a claim can be true or false. "5 + 3 = 8" is a statement.

Mistake: Confusing "sometimes true" with true statements

Why: "It is raining" can be true today and false tomorrow: its truth value depends on when and where you say it.

Correct: A math statement like "$2 + 2 = 4$" does not depend on time or place: it is ALWAYS true. For a sentence like "It is raining", first say when and where you mean.

Mistake: Thinking false means "bad" or "wrong to say"

Why: False simply means the statement does not match reality. It is not a judgment about the person.

Correct: False is a logical classification. "$3 > 5$" is false because 3 is not greater than 5.

Fact-Checking Information

When you read news or information online, you evaluate whether statements are true or false.

If someone says "The Earth is the largest planet in our solar system," you can check: Is this true or false? (It is false - Jupiter is the largest.)

Testing Answers in Math

When you solve a math problem, you can check your answer by substituting it back.

If you solve $x + 5 = 12$ and get $x = 7$, check: Is "$7 + 5 = 12$" true? Yes! Your answer is correct.

Computer Programs

Computers use true/false logic constantly to make decisions.

A game checks: "Is the player's score greater than the high score?" If TRUE, save new high score. If FALSE, keep the old one.

A statement is a sentence that is either true or false, but not both

True statements are correct and match mathematical rules or reality

False statements are incorrect and do not match reality

Questions and commands are NOT statements

In math, true statements are ALWAYS true, not just sometimes

Q: Can a statement be both true and false?

A: No! In logic, every statement has exactly one of the two truth values: it is true OR false, never both and never something in between. This rule is called the principle of bivalence ("two values").

Q: Is "I like pizza" a statement?

A: This is tricky! It is a statement (it can be true or false), but its truth depends on the person saying it. In math, we prefer statements with clear, objective truth values like "$5 > 3$".

Q: What about statements whose truth we do not know yet?

A: Even if we do not know whether something is true, it still has a truth value. "There are exactly 100 grains of sand on this beach" is either true or false - we just might not know which.

Exit Presentation

True and False Statements

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Logic & Sets / Boolean Logic

True and False Statements

Learn to identify and evaluate statements as true or false.

Elementary15 min

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