Free preview 2/11

Question 1
Which of the following best describes the closure property?
No clue? Show me the answer
Correct answerCorrect!Incorrect
BPerforming an operation on two numbers from a set always produces a result in the same set
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Recall the three properties
The three properties are closure, commutative, and associative. We need to match the definition given to the correct property.
Step 2: Match the definition to closure
Closure means: when you apply an operation to any two numbers in a set, the result is still in that same set. For example, $3 + 5 = 8$ — all three numbers are whole numbers.
Step 3: Choose the correct answer
The option 'Performing an operation on two numbers from a set always produces a result in the same set' matches the definition of the closure property exactly.
Method #2Process of Elimination
Step 1: Identify what is being asked
We need the definition of the closure property from the four options provided.
Step 2: Eliminate 'changing the order'
'Changing the order of numbers does not change the result' describes the commutative property, not closure. Eliminate this option.
Step 3: Eliminate 'changing the grouping'
'Changing the grouping of numbers does not change the result' describes the associative property. Eliminate this option.
Step 4: Eliminate 'a special number leaves others unchanged'
'A special number leaves other numbers unchanged under an operation' describes an identity element (like 0 for addition or 1 for multiplication). Eliminate this option.
Step 5: Select the remaining option
The only remaining option — 'Performing an operation on two numbers from a set always produces a result in the same set' — correctly defines closure.
Question 2
A student claims that the set of whole numbers is closed under subtraction. Which example proves this claim is FALSE?
No clue? Show me the answer
Correct answerCorrect!Incorrect
C $2 - 9 = - 7$
Step-by-step walkthrough
Choose a solution method
Method #1Direct Approach
Step 1: Understand what we need
To disprove closure, we need a counterexample: two whole numbers whose difference is not a whole number. Whole numbers are $0, 1, 2, 3, \dots$ — they do not include negatives.
Step 2: Check each option
We need to find a subtraction of two whole numbers that gives a result outside the set of whole numbers (i.e., a negative result).
Step 3: Identify the counterexample
$2 - 9 = - 7$ . Here, both 2 and 9 are whole numbers, but $- 7$ is negative — it is not in the set of whole numbers. This is a valid counterexample that disproves the claim.
Method #2Process of Elimination
Step 1: What counts as a counterexample?
A counterexample must show two whole numbers that, when subtracted, give a result that is not a whole number (i.e., a negative number or a fraction).
Step 2: Eliminate $5 - 5 = 0$
$5 - 5 = 0$ , and 0 is a whole number. This does not disprove closure. Eliminate this option.
Step 3: Eliminate $8 - 3 = 5$
$8 - 3 = 5$ , and 5 is a whole number. This does not disprove closure. Eliminate this option.
Step 4: Eliminate $10 - 0 = 10$
$10 - 0 = 10$ , and 10 is a whole number. This does not disprove closure. Eliminate this option.
Step 5: Select the correct counterexample
$2 - 9 = - 7$ . The result $- 7$ is negative, so it is not a whole number. This is the counterexample that disproves the claim.

Free preview

9 more questions in this topic

Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions. Your answers here are kept on this device.

Step-by-step walkthrough

Step 1: Recall the three properties

Step 2: Match the definition to closure

Step 3: Choose the correct answer

Step 1: Identify what is being asked

Step 2: Eliminate 'changing the order'

Step 3: Eliminate 'changing the grouping'

Step 4: Eliminate 'a special number leaves others unchanged'

Step 5: Select the remaining option

Step-by-step walkthrough

Step 1: Understand what we need

Step 2: Check each option

Step 3: Identify the counterexample

Step 1: What counts as a counterexample?

Step 2: Eliminate $5 - 5 = 0$

Step 3: Eliminate $8 - 3 = 5$

Step 4: Eliminate $10 - 0 = 10$

Step 5: Select the correct counterexample

9 more questions in this topic