2.1.3 Write each of the following statements as a precise mathematical statement
ID: 3871534 • Letter: 2
Question
2.1.3
Write each of the following statements as a precise mathematical statement. Use variable names to denote arbitrary numbers in the domain. Your statements should avoid mathematical terms (such as "square root" or "multiplicative inverse") and should be expressed using algebra.
(a) The square root of every positive number less than one is greater than the number itself.
(c) Every real number besides 0 has a multiplicative inverse.
(d) Among every consecutive three integers, there is a multiple of 3.
Note: a number is a multiple of 3 if it can be expressed as 3k for some integer k. Avoid the use of the word "consecutive" in your statement.
Please answer all questions
Explanation / Answer
Hi,
Before answering, lets understand few symols,
is used to represent 'for all' ,
is used to represent 'there exists atleast one'
is used for 'belongs to'
let x be the variable
set of integer is represented by Z. and set of real numbers by R.
Multiplicative inverse of a number is the number with when multiplied gives 1, hence multiplicative inverse of x is 1/x or x1
- this is logical AND statement used to combine two conditions
v - this is logica OR
-> is used for implication, i.e to represent if A then B by A->B
now, using the above symbol definitions we can write the given statements in algebra
a. x Z x<1 -> x >x
b. x y((xRx0) -> xy=1(definition of multiplicative inverse)
c. xyz (x,y,zZx=y-1 z=y+1) -> x%3= 0 v y%3=0 v z%3=0 this is one way of writing it, for all x,y,z where x=y-1 and z=y+1 i.e they are consevutive, one of them will divide 3 i.e %3 is 0 ,
one more way is expressing in terms of 3k
xyz k(x,y,z,kZx=y-1 z=y+1) -> x=3k v y=3k v z=3k
there exists a k which is also an integer, such that number divides 3.
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