Check all the statements that are true: There are n^m functions from a set of n
ID: 3012912 • Letter: C
Question
Check all the statements that are true: There are n^m functions from a set of n elements to a set of m elements. If a task can be done either in one of n ways or in one of m ways, then there are n+m ways to do the task. Combinations C(n, r) are symmetrical in r with respect to the point r = n/2. A finite set with n members has C(n, k) subsets of size k. If n and k are positive integers with n greaterthanorequalto k, then C(n+1, k) = C(n, k) + C(n, k+1). If n is a nonnegative integer, then the sum of all the C(n, k) is 2^n. There are n! bijections from a set with n elements to itself. If there are 2n+1 objects in n boxes, then at least one box must contain at least 3 objects. If a procedure can be broken down into a sequence of two tasks, and if there are n ways to do the first task, and m ways to do the second task, then there are nm ways to do the procedure. If n and r are nonnegative integers and r lessthanorequalto n, then C(n, r) = P(n, r)/r!. The cardinality of a cartesian product of sets is the product of the cardinalities of the individual sets. If S is a finite set, S has 2|S| subsets. If n is a nonnegative integer, then the alternating sums of all the C(n, k) are 0. A surjective function from a set of n elements to a set of n elements is automatically injective. An injective function from a set of n elements to a set of n elements is automatically surjective.Explanation / Answer
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A) FALSE
There shall be m^n sets. For each element of set A we have m no of options
for n elements we have m options
i.e. m X m X m X m X m X n times
= m^n
B) FALSE
Since it has not been mentioned that m and n sets are mutually excusie. Hence it cannot be said that the total number of ways of doing the task is m+n.
The total number of ways of doing the task is m + n - m&n( Common ways in set m and set n)
If m and n are mututally exclusive then the given statement can be considered as truth.
C) Combinations are symmetric about n = r/2
TRUE
Since nCr = nC (n-r) By combinations property
D) Number of subset = nCk
TRUE
Number of ways of selection of k elements from given set of n elements is nCk
Hence nCk determines the number of subset possible.
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'=>for r = n/2 the value of combinations are symmetric.
D)