Introduction:
In transistor circuits, there are two common configurations: common base (CB) and common emitter (CE). The current gain in the CB configuration is denoted by 'a', while the current gain in the CE configuration is denoted by 'b'. We are given that x = 1/a and y = 1/b.

Analysis:

1. XY = 1:
To prove this statement, we need to substitute the given values of x and y into the equation and simplify it.

x = 1/a and y = 1/b

Substituting these values into the equation XY:

XY = (1/a)(1/b) = 1/(ab)

Since 'a' and 'b' are current gains, they are always positive values. Therefore, ab is also a positive value. Hence, 1/(ab) is equal to 1.

Therefore, XY = 1.

2. X - Y = 1:
Similarly, we can substitute the given values of x and y into the equation and simplify it.

x = 1/a and y = 1/b

Substituting these values into the equation X - Y:

X - Y = (1/a) - (1/b) = (b - a)/(ab)

Here, 'a' and 'b' are always positive values. Therefore, (b - a) is either positive or negative, but not equal to zero. Hence, (b - a)/(ab) is not equal to 1.

Therefore, X - Y ≠ 1.

3. 2X = 1 - Y:
Again, substitute the given values of x and y into the equation and simplify it.

x = 1/a and y = 1/b

Substituting these values into the equation 2X:

2X = 2(1/a) = 2/a

Also, substitute the value of y into the equation 1 - Y:

1 - Y = 1 - (1/b) = (b - 1)/b

Comparing both equations, we can see that 2/a ≠ (b - 1)/b.

Therefore, 2X ≠ 1 - Y.

4. XY = 0:
To prove this statement, we need to substitute the given values of x and y into the equation and simplify it.

x = 1/a and y = 1/b

Substituting these values into the equation XY:

XY = (1/a)(1/b) = 1/(ab)

If either 'a' or 'b' is equal to zero, then ab = 0 and 1/(ab) is undefined.

Hence, XY ≠ 0.

Conclusion:
From the analysis above, we can conclude the following:
1. XY = 1
2. X - Y ≠ 1
3. 2X ≠ 1 - Y
4. XY ≠ 0

X-Y=1