Theorem 3.7: a) Let f be a function from A to B, Inverse relation f-Iis a function from B to A if only if f is one to one 6(b) Let f be an everywhere function from A to B, Inverse relation f- is an everywhere function from b to A if only if f is one-to-one correspondence
❖Theorem 3.7:(a) Let f be a function from A to B, Inverse relation f -1 is a function from B to A if only if f is one to one ❖(b) Let f be an everywhere function from A to B, Inverse relation f -1 is an everywhere function from B to A if only if f is one-to-one correspondence
4.1.2 Pigeonhole principle: Strong Form 今 Theoren4.2:Letq1,q2…, gn be positive integers. If q1+q2+.+qn-n+l objects are put into n boxes, then either the first box contains at least qi objects, or the second box contains at least q2 objects, .., or the nth box contains at least qn objects. 今 Proof: Suppose that we distribute 91+q2+.+qn-n+l objects among n boxes
4.1.2 Pigeonhole principle:Strong Form ❖ Theorem 4.2: Let q1 ,q2 ,…,qn be positive integers. If q1+q2+…+qn -n+1 objects are put into n boxes, then either the first box contains at least q1 objects, or the second box contains at least q2 objects, … , or the nth box contains at least qn objects. ❖ Proof:Suppose that we distribute q1+q2+…+qn -n+1 objects among n boxes
)If n(r-1)+l objects are put into n boxes, then at least one of the boxes contains r or more of the objects. Equivalently, (2)If the average of n non-negative integers mi, mose.,m is greater than r-1: (m,+m2+.+mm/n>r-1, then at least one of the integers is greater than or equal to r 今 Proof:(1)q1=q2=…=qn=r q1+q2+.+qn-n+l=rn-n+=(r-1)n+l, then at least one of the boxes contains r or more of the objects (2)(m1+m2+…+mn)>(r-1)n, 今(m1+m2+…+mn)≥(r-1n+1
❖ (1)If n(r-1)+1 objects are put into n boxes, then at least one of the boxes contains r or more of the objects. Equivalently, ❖ (2)If the average of n non-negative integers m1 ,m2 ,…,mn is greater than r-1: (m1+m2+…+mn )/n>r-1, then at least one of the integers is greater than or equal to r. ❖ Proof:(1)q1=q2=…=qn=r ❖ q1+q2+…+qn -n+1=rn-n+1=(r-1)n+1, then at least one of the boxes contains r or more of the objects。 ❖ (2)(m1+m2+…+mn )>(r-1)n， ❖ (m1+m2+…+mn )≥(r-1)n +1
Example 6: Two disks, one smalle her than the other, are each divided into 200 congruent sectors. In the larger disk 100 of the sectors are chosen arbitrarily and painted red; the other 100 of the sectors are painted blue. In the smaller disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The small disk is then placed on the larger disk so that their centers coincide. Show that it is possible to align the two disks so that the number of sectors of the small disk whose color matches the corresponding sector of the large disk is at least 100 &o if the large disk is fixed in place g there are 200 possible positions for the small disk such that each sector of the small disk is contained in a sector of the large disk g color matches the corresponding 令20000/200=100>100-1 Position with at least 100 color matches
❖ Example 6:Two disks, one smaller than the other, are each divided into 200 congruent sectors. In the larger disk 100 of the sectors are chosen arbitrarily and painted red; the other 100 of the sectors are painted blue. In the smaller disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The small disk is then placed on the larger disk so that their centers coincide. Show that it is possible to align the two disks so that the number of sectors of the small disk whose color matches the corresponding sector of the large disk is at least 100. ❖ if the large disk is fixed in place ❖ there are 200 possible positions for the small disk such that each sector of the small disk is contained in a sector of the large disk. ❖ color matches the corresponding ❖ 20000/200=100>100-1 ❖ Position with at least 100 color matches
4.2 Permutations of sets 842.1 Basic counting principles Theorem 4.3(Addition principle): IfA, A2, .. A are disjoint sets, then the number of elements in the union of these sets is the sum of the numbers of elements in them 令|A1UA20…∪An=A1+A2+.+n
4.2 Permutations of sets ❖4.2.1 Basic counting principles ❖ Theorem 4.3(Addition principle): If A1 , A2 , … , An are disjoint sets, then the number of elements in the union of these sets is the sum of the numbers of elements in them. ❖ | A1∪A2∪…∪An |=|A1 |+|A2 |+…+|An |