ary digit binary string or a binary sequence of length sequen pits. he examples are 00000000, 01010101, and 01001011. The bit in position 0 is the furthest bit to the left, in 01111111, the 0 bit is in position 0. In 11011111 the 0 bit is in position 2. binary string 1110000110 is a binary string of length 10 which contains the substring 0000 starting at tion 3. sider all length 14 binary strings. Let A; be the collection of all such strings in which the length 4 string 1000 occurs starting at position 1. How many of A₁ will be non-empty? 11 For non-empty A₁, what is A₁? 11 X How many of the intersections A₂A, with i j will be non-empty? For a non-empty A₁ A₁ with i ‡j, what is A₁ A₁? How many of the intersections An An Ak with i, j, k distinct will be non-empty? For a non-empty A₁ A₁ Ak with i,j,k distinct what is A₁ A₁ Ak? (If all triple intersections are oty, answer 0.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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b,c, and d

A bit or a binary digit is a 0 or a 1. A binary string or a binary sequence of length 8 is a sequence of 8 bits.
Some examples are 00000000, 01010101, and 01001011. The bit in position 0 is the furthest bit to the left,
e.g. in 01111111, the 0 bit is in position 0. In 11011111 the 0 bit is in position 2.
The binary string 1110000110 is a binary string of length 10 which contains the substring 0000 starting at
position 3.
Consider all length 14 binary strings. Let A; be the collection of all such strings in which the length 4
substring 1000 occurs starting at position 1.
(a) How many of A₁ will be non-empty? 11
(b) For non-empty A₁, what is A₁? 11
X
(c) How many of the intersections A₁ Aj with i j will be non-empty?
(d) For a non-empty A₁ A₁ with i ‡j, what is A₁ A₁?
(e) How many of the intersections A₁ Ajn A with i,j,k distinct will be non-empty?
(f) For a non-empty Ain Ajn Ak with i, j, k distinct what is Ain Ajn Ak? (If all triple intersections are
empty, answer 0.)
(g) How many binary strings of length 14 contain the substring 1000?
Transcribed Image Text:A bit or a binary digit is a 0 or a 1. A binary string or a binary sequence of length 8 is a sequence of 8 bits. Some examples are 00000000, 01010101, and 01001011. The bit in position 0 is the furthest bit to the left, e.g. in 01111111, the 0 bit is in position 0. In 11011111 the 0 bit is in position 2. The binary string 1110000110 is a binary string of length 10 which contains the substring 0000 starting at position 3. Consider all length 14 binary strings. Let A; be the collection of all such strings in which the length 4 substring 1000 occurs starting at position 1. (a) How many of A₁ will be non-empty? 11 (b) For non-empty A₁, what is A₁? 11 X (c) How many of the intersections A₁ Aj with i j will be non-empty? (d) For a non-empty A₁ A₁ with i ‡j, what is A₁ A₁? (e) How many of the intersections A₁ Ajn A with i,j,k distinct will be non-empty? (f) For a non-empty Ain Ajn Ak with i, j, k distinct what is Ain Ajn Ak? (If all triple intersections are empty, answer 0.) (g) How many binary strings of length 14 contain the substring 1000?
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