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olya-2409 [2.1K]
2 years ago
8

Weak Induction

Computers and Technology
1 answer:
slega [8]2 years ago
7 0

Answer:

Following are the answer to this question:

Explanation:

In option 1:

The value of n is= 7, which is (base case)

\to 3^7

when n=k for the true condition:

\to 3^k

when n=k+1 it tests the value:

\to 3^{(k+1)}= 3^k,3\\\to < (k!) 3 \ substituting \ equation \\\to

since k>6  hence the value is KH>3 hence proved.

In option 2:

when:

for n=1:(base case)

\log(1!)

0<=0 \\ condition is true

when the above statement holds value n=1

when n=k

\log(k!)

when n=k+1

\log(k+1)!=\log(k!)+\log(k+1)\\

             

\because k \log k      [\therefore KH>K \Rightarrow  \log(KH>\loK)]

In option 3:

when n=1:

A_1 \cup B=A_1 \cup B\\\\

when n=k

\to (A_1\cap A_2 \cap.....A_k) \cup B\\=(A_1\cup B) \cap(A_2\cup B_2)....(A_k \capB).....(a)\\\to n= k+1\\ \to (A_1\cap A_2 \cap.....A_{kH}) \cup B= (A_1\cup B)\\\\\to  [(A_1\cap A_2 \cap.....A_{k}) \cup B]\cap (A_{KH}\cup B)\\\\\to  [(A_1\cup B) \cap (A_2 \cup B) \cap (A_3\cup B).....(A_k\cup B)\cap (A_{k+1} \cup B)\\\\  \ \ \ \ \ \ substituting \ equation \ a \\\\

hence n=k+1 is true.

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