## Q-2: [4+4 marks ] a) Define A△B=(A−B)∪(B−A) , where A,B are any two sets. Show that (AΔB)ΔC=AΔ(BΔC) b) Prove or disprove: i. ⌈x⌉=−⌊x⌋ . ii. For all positive real numbers x and y,⌊x⋅y⌋≤⌊x⌋⋅⌊y⌋ .

EXPERT ANSWER

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# Other Math

## Determine whether each of the following is true or false. If true, prove the statement. If false, find a counterexample.

## Write the function in the form f (z) = u (x,y) + iv (x,y) and find all points where the Cauchy-Riemann equations are satisfied

## Suppose our stream consists of the integers 3, 1, 4, 1, 5, 9, 2, 6, 5. Our hash functions will all be of the form h(x) = ax+b mod 32 for some a and b. You should treat the result as a 5-bit binary integer. Determine the tail length for each stream element and the resulting estimate of the number of distinct elements if the hash function is

## Consider the pattern P=a3ba. Construct the table and the corresponding labeled directed graph used in the “fast” pattern matching algorithm

## 6) Prove that gcd(a,b)×gcd(a,c)×bcd(b,c)≥(gcd(a,b,c)) 3 for all positive integers, a,b , and c . Intuitively, without proof (though if you can provide a proof that would be great), determine the situations where the two sides of this equation are equal. (Note: The originally posed question is relatively easy, so be looking for straight-forward observations about the property of the ged function as opposed to something detailed and esoteric.)

Determine whether each of the following is true or false. If true, prove the statement. If false, find a counterexample. a) For all integers a, b,and c, if a | b and a | c then a | (3b − 5c). b) For all integers a ≥ 4 and b, if a | 3b then …

Write the function in the form f (z) = u (x,y) + iv (x,y) and find all points where the Cauchy-Riemann equations are satisfied EXPERT ANSWER

Suppose our stream consists of the integers 3, 1, 4, 1, 5, 9, 2, 6, 5. Our hash functions will all be of the form h(x) = ax+b mod 32 for some a and b. You should treat the result as a 5-bit binary integer. Determine the tail length for each stream element and the …

Consider the pattern P=a3ba. Construct the table and the corresponding labeled directed graph used in the “fast” pattern matching algorithm EXPERT ANSWER

EXPERT ANSWER Let gcd(a,b,c) = m.Therefore, m divides a, b and c.Then, gcd(a,b) = mx, gcd(b,c) = my, gcd(a,c) = mz where x, y, z are all greater than or equal to 1. Thus, gcd(a,b)xgcd(b,c)x gcd(a,c) = mxmymz = xyzm3 And [gcd(a,b,c)]3 = m3. xyzm3 is greater than or equal to m3 Therefore, gcd(a,b)xgcd(b,c)x gcd(a,c) …