1 State whether the following statements are true or false 9

1. State whether the following statements are true or false: 9) a) Two sets are equal if and only if they have the same number of elements. (b) If A is a proper subset of B then the cardinality of A and B are same. (c) If two sets are disjoint then their intersection is not an empty set (d) If range and codomain of a function f are equal then f is a surjective function. (e) If f is not bijective then f can not be invertible (f) In an algorithm, infinite number of steps may be used to get the desired output. (g) If a 24-hour clock read 18:00, then after 95 hours it will read 17:00 (h) Every prime number is not even (i) The integers 12, 17 and 21 are pairwise relatively prime number Page 1 of3 Please go on to the next page

Solution

I am solving the Question 1 with 9 sub-parts, please post multiple questions to get the remaining problems. Thanks

1)

a) The statement is FALSE since the sets will be equal if and only if the two sets have the same elements not the same number of elements

Counter Examples

Set A = {1,2,3}
Set B = {4,5,6}

The number of elements in both the sets are same but they are not equal

b)

The statement is TRUE since if the set is a proper subset then it must have the same cardinality

c)

The statement is FALSE

If two sets are disjoint, then there intersection will contain 0 elements or null elements since they are disjoing and doesn\'t share a common element

d) The staetement is TRUE

since if codomain and range are same, then it means the set has all the inputs in the domain otherwise it would not have the complete codomain, hence the set is surjective

e) The statement is FALSE

since the function can be injective but not bijective at the same time

f) The statement is FALSE

g)

The statement is TRUE

After multiples of 24 hour, it will read the same time

24 * 4 = 96, hence after 96 hours it will be again at 18:00 but at 95 hours, the clock will be at 17:00

h)

The statement is FALSE

Since 2 is a prime number as well as an even number

i)

The statement is FALSE

Since the set {12,21} are not relatively prime since they share a common factor of 3, the numbers are said to be relatively prime if the gcd of them is 1 but in this case gcd will be equal to 3