## Functions

Question 1 |

How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set?

2 ^{n} | |

2 ^{n}-1 | |

2 ^{n}-2 | |

2(2 ^{n}– 2) |

Question 1 Explanation:

The number of onto functions from set of m elements to set of n elements, if m>n is

n

i.e., 2

If there are 'm' elements in set A, 'n' elements in set B then

The number of functions are : n

The number of injective or one-one functions are

The number of surjective functions are:

If m
If m>n, then n! *

Given that m=n, n=2

2! *

n

^{m}– (2^{n}– 2)i.e., 2

^{n}– (2^{2}– 2) = 2^{n}– 2If there are 'm' elements in set A, 'n' elements in set B then

The number of functions are : n

^{m}The number of injective or one-one functions are

^{n}P_{m}The number of surjective functions are:

If m

^{m}C

_{n}

Given that m=n, n=2

2! *

^{n}C

_{2}

Question 3 |

On the set N of non-negative integers, the binary operation __________ is
associative and non-commutative.

fog |

Question 3 Explanation:

The most important associative operation thats not commutative is function composition. If you have two functions f and g, their composition, usually denoted fog, is defined by

(fog)(x) = f(g(x))

It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.

Note that if fog exists then gof might not even exists.

(fog)(x) = f(g(x))

It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.

Note that if fog exists then gof might not even exists.

Question 4 |

The function f(x,y) = x

^{2}y - 3xy + 2y + x hasno local extremum | |

one local minimum but no local maximum | |

one local maximum but no local minimum | |

one local minimum and one local maximum |

Question 4 Explanation:

Note: Out of syllabus.

Question 6 |

Let A and B be sets with cardinalities m and n respectively. The number of one-one mappings (injections) from A to B, when m < n, is:

m ^{n} | |

^{n}P_{m} | |

^{m}C_{n} | |

^{n}C_{m} |

Question 6 Explanation:

Let,

A one-one function 'f' assigns each element a

i.e.,

A one-one function 'f' assigns each element a

_{i}of A a distinct element, b_{j}=f(a_{i}) of B_{i}for a, there are n choices, for a_{2}there are n-1 choices, for a_{m}there are (n-(m-1)) choices.i.e.,

There are 6 questions to complete.