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Marta_Voda [28]
3 years ago
12

2.5 Refer to Exercise 2.4. Use the identities A = A ∩ S and S = B ∪ B and a distributive law to prove that a A=(A∩B)∪(A∩B). b If

B⊂AthenA=B∪(A∩B). c Further, show that ( A ∩ B ) and ( A ∩ B ) are mutually exclusive and therefore that A is the union of two mutually exclusive sets, (A ∩ B) and (A ∩ B). d AlsoshowthatBand(A∩B)aremutuallyexclusiveandifB⊂A,Aistheunionoftwo mutually exclusive sets, B and (A ∩ B)

Mathematics
1 answer:
natka813 [3]3 years ago
5 0

Answer:

All identities proved

Step-by-step explanation:

The step by step proving using the distributive law property is as shown in the attachment.

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Use the principle of inclusion and exclusion to find the number of positive integers less than 1,000,000 that are not divisable
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This is a simple problem based on combinatorics which can be easily tackled by using inclusion-exclusion principle.
We are asked to find number of positive integers less than 1,000,000 that are not divisible by 6 or 4.
let n be the number of positive integers.
∴ 1≤n≤999,999
Let c₁ be the set of numbers divisible by 6 and c₂ be the set of numbers divisible by 4.
Let N(c₁) be the number of elements in set c₁ and N(c₂) be the number of elements in set c₂.

∴N(c₁) = \frac{999,999}{6} = 166666
N(c₂) = \frac{999,999}{4} = 250000
∴N(c₁c₂) = \frac{999,999}{24} = 41667
∴ Number of positive integers that are not divisible by 4 or 6,

N(c₁`c₂`) = 999,999 - (166666+250000) + 41667 = 625000
Therefore, 625000 integers are not divisible by 6 or 4
8 0
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Answer:

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