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Questions and Answers
Which law states that a + b = z?
Which law states that a + b = z?
The commutative law for multiplication states that a x b = b x a.
The commutative law for multiplication states that a x b = b x a.
True
What does the associative law for addition state?
What does the associative law for addition state?
a + (b + c) = (a + b) + c
Which of the following expressions represents the distributive law?
Which of the following expressions represents the distributive law?
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What is the reflexive property?
What is the reflexive property?
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What does transitive property state?
What does transitive property state?
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What does the inequality symbol 'a > b' mean?
What does the inequality symbol 'a > b' mean?
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If x < 1 or x > 5, then x can be a value between 1 and 5.
If x < 1 or x > 5, then x can be a value between 1 and 5.
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What is the theorem related to adding the same number to both sides of an inequality?
What is the theorem related to adding the same number to both sides of an inequality?
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What can be concluded from multiplying both sides of an inequality by the same positive number?
What can be concluded from multiplying both sides of an inequality by the same positive number?
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Study Notes
Basic Laws of Natural Numbers
- Law of Closure for Addition: If a and b are natural numbers, then their sum a + b is also a natural number. Example: 2 + 5 = 7.
- Commutative Law for Addition: The order of addition does not affect the sum, a + b = b + a. Example: 5 + 7 = 7 + 5 = 12.
- Associative Law for Addition: Grouping does not affect the sum, a + (b + c) = (a + b) + c. Example: 12 + (13 + 22) = (12 + 13) + 22 = 47.
- Law of Closure for Multiplication: If a and b are natural numbers, then their product a × b is also a natural number. Example: 2 × 4 = 8.
- Commutative Law for Multiplication: The order of multiplication does not affect the product, a × b = b × a. Example: 12 × -3 = -3 × 12 = -36.
- Associative Law for Multiplication: Grouping does not affect the product, a(b × c) = (a × b) × c. Example: 6 × (-4 × -3) = (6 × -4) × -3 = 72.
- Distributive Law: This law links addition and multiplication, a(b + c) = ab + ac. Example: 3(5 + 8) = 15 + 24 = 39; 3(5 - 8) = 15 - 24 = -9.
Basic Laws of Equality
- Reflexive Property: Any quantity is equal to itself, a = a.
- Symmetric Property: If a = b, then b = a.
- Transitive Property: If a = b and b = c, then a = c; things equal to the same thing are equal to each other.
- Addition of Equals: If a = b and c = d, then a + c = b + d.
- Multiplication of Equals: If a = b and c = d, then ac = bd.
Inequalities
- Definition: An inequality expresses a relationship between two quantities, indicating one is greater or less than the other using symbols like >, <, ≥, and ≤.
- Compound Inequalities: Can indicate a range of values using "and" (e.g., 1 < x ≤ 5 means x is between 1 and 5) or "or" (e.g., x < 1 or x > 5 means x is outside the interval (1, 5)).
Theorems of Inequalities
- Theorem 1: Adding the same number to both sides of an inequality will not change the sense of the inequality. (If a > b, then a + c > b + c).
- Theorem 2: Multiplying both sides of an inequality by a positive number preserves the inequality. (If a > b and c > 0, then ac > bc). Example: For 8 > -5, multiplying both sides by 3 gives 24 > -15.
- Division of Inequalities: Similar to multiplication, dividing by a positive number maintains the inequality's sense. (If a > b and c > 0, then a/c > b/c).
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Description
This quiz covers the basic laws of natural numbers, focusing on the laws of closure, commutative, and associative properties for addition. It provides examples to illustrate each law, helping students to grasp these fundamental concepts. Perfect for anyone looking to solidify their understanding of basic arithmetic operations.
