Which pair(s) of expressions are equal in value? Check all that apply.
[tex]$\lfloor 4.9\rfloor$[/tex] and [tex]$\lceil 3.1\rceil$[/tex]
[tex]$\lfloor 15.2\rfloor$[/tex] and [tex]$\lfloor 14.8\rfloor$[/tex]
[tex]$\lfloor-6\rfloor$[/tex] and [tex]$\lceil-6\rceil$[/tex]
[tex]$\lceil-3.2\rceil$[/tex] and [tex]$\lceil-2.6\rceil$[/tex]
Answer (1)
Evaluate each expression using the definitions of floor and ceiling functions.
Compare the values of each pair of expressions.
Identify that ⌊ 4.9 ⌋ = 4 and ⌈ 3.1 ⌉ = 4 , so they are equal.
Identify that ⌊ − 6 ⌋ = − 6 and ⌈ − 6 ⌉ = − 6 , so they are equal. ⌊ 4.9 ⌋ and ⌈ 3.1 ⌉ , ⌊ − 6 ⌋ and ⌈ − 6 ⌉
Explanation
Understanding Floor and Ceiling Functions We need to evaluate each floor and ceiling expression and then compare the pairs to see which ones are equal. The floor function, denoted by ⌊ x ⌋ , gives the greatest integer less than or equal to x . The ceiling function, denoted by ⌈ x ⌉ , gives the smallest integer greater than or equal to x .
Evaluating the Expressions Let's evaluate each expression:
⌊ 4.9 ⌋ = 4 (The greatest integer less than or equal to 4.9 is 4).
⌈ 3.1 ⌉ = 4 (The smallest integer greater than or equal to 3.1 is 4).
⌊ 15.2 ⌋ = 15 (The greatest integer less than or equal to 15.2 is 15).
⌊ 14.8 ⌋ = 14 (The greatest integer less than or equal to 14.8 is 14).
⌊ − 6 ⌋ = − 6 (The greatest integer less than or equal to -6 is -6).
⌈ − 6 ⌉ = − 6 (The smallest integer greater than or equal to -6 is -6).
⌈ − 3.2 ⌉ = − 3 (The smallest integer greater than or equal to -3.2 is -3).
⌈ − 2.6 ⌉ = − 2 (The smallest integer greater than or equal to -2.6 is -2).
Comparing the Pairs Now, let's compare the pairs:
⌊ 4.9 ⌋ = 4 and ⌈ 3.1 ⌉ = 4 . These are equal.
⌊ 15.2 ⌋ = 15 and ⌊ 14.8 ⌋ = 14 . These are not equal.
⌊ − 6 ⌋ = − 6 and ⌈ − 6 ⌉ = − 6 . These are equal.
⌈ − 3.2 ⌉ = − 3 and ⌈ − 2.6 ⌉ = − 2 . These are not equal.
Identifying Equal Pairs Therefore, the pairs of expressions that are equal in value are ⌊ 4.9 ⌋ and ⌈ 3.1 ⌉ , and ⌊ − 6 ⌋ and ⌈ − 6 ⌉ .
Final Answer The pairs of expressions that are equal are ⌊ 4.9 ⌋ and ⌈ 3.1 ⌉ , and ⌊ − 6 ⌋ and ⌈ − 6 ⌉ .
Examples
Floor and ceiling functions are used in computer science for memory allocation, where you might need to round up the size of data to the nearest whole number of bytes (ceiling function) or determine how many full pages of memory are needed (floor function). They also appear in everyday scenarios like calculating the number of whole items you can buy with a certain amount of money or determining the minimum number of containers needed to store a specific quantity of items. Understanding these functions helps in resource optimization and decision-making in various practical situations. For example, if you have $10.50 and each apple costs 2 , t h e f l oor f u n c t i o nh e lp syo u d e t er min e t ha t yo u c anb u y \lfloor \frac{10.50}{2} \rfloor = \lfloor 5.25 \rfloor = 5$ apples.
