It is one question
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(b) p Λ (q V r) becomes False Λ (False V True) which is False
(p Λ q) V r becomes (False Λ False) V True which is True
Since p Λ (q V r) is not equivalent to (p Λ q) V r in this situation, they are not equivalent.
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(b) Consider x = 1. Then 1 ≥ y + 1 , which means y ≤ 0, which contradicts the premise that y is a positive integer.
(c) x is a positive integer larger than 1, y is a positive integer
(d) There exists an x value for which there is not a y such that x ≤ y + 1, where x and y are positive integers.
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|x| = x
|y| = y
x + y ≥ 0 → |x + y| = x + y
|x + y| = |x| + |y|
|y| = -y
If x + y ≥ 0 → |x + y| = x + y < |x| + |y| = x - y
If x + y < 0 → |x + y| = -x - y ≤ |x| + |y| = x - y
Replace y with x and x with y. This is proved in case (ii).
|x| = -x
x + y = -(-x + -y)
|x + y| = -(x + y) = -x + -y = |x| + |y|
For n = 1,
1/(1(2)) = 1/2 = 1 - 1/(1+1)
Assume the equation holds for n = k.
For n = k+1,
1 / 1(2) + 1 / 2(3) + 1 / (k-1)k + 1 / k(k+1) = 1 - 1/k + 1 / k(k+1) = 1 - (k+1) / k(k+1) + 1 / k(k+1) = 1 - k / k(k+1) = 1 - 1 / (k+1)
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