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The type of logic gate that the output X represent is NAND.

1. XY’ + XY = X
By proof of perfect induction X = 0, Y =1
XY’ + XY = X
LHS = 0.0 + 0.1 = 0 + 0 = 0
By proof of perfect induction X = 1, Y = 0
LHS = 1.1 + 1.0 = 1 + 0 = 1.

2. X + Y = Y + X
By proof of perfect induction X = 0, Y = 1
LHS = 0 + 1 = 1
RHS = 1 + 0=1
LHS = RHS
By proof of perfect induction X = 1, Y = 0
LHS = 1 + 0 = 1
RHS = 0 + 1 = 1
LHS = RHS

The dual form of

1. 0.X + X.Y + 1.X   is 1 + X’. X’ + Y’. 0 + X’
2. X.(Y+Z) = X.Y + X.Z  is  X’ + (Y’.Z’) = X’ + Y’. X’ + Z’
3. X + X’.Y = X + Y  is  X’. X + Y’ = X’. Y’
4. 1 + X = 1  is  0.X = X.

1. (A + B).C
= AC + BC.

2. (A + BC). (D + EF)
(A + BC) . D + (A + BC). EF
AD + BCD + AEF + BCEF.

1. De Morgan’s theorem 1:
When the OR sum of two variables is inverted, this is the same as inverting each variable individually and then ANDing these inverted variables.
(X + Y)’ = X’. Y’

2. De Morgan’s theorem 2:
When the AND product of two variables is inverted, this is the same as inverting each variable individually and then ORing them.
(X.Y)’ = X’ + Y’

The principle of duality theorems states that starting with a Boolean relation another Boolean relation can be derived by.

1. Changing each OR sign (+) to and AND sign (.)
2. Changing each AND sign (.) to an OR sign (+)
3. Changing each 0 by 1 and each 1 by 0.

The three basic logic gates are NOT, OR and AND gate.

The different universal gates are NAND and NOR.

A gate which can be used to create any Logic gate is called Universal Gate.

A table that represents all possible values of logical variables along with all the possible results of the given combinations of values.

The three logic operations are Nor, Or and And.

The NOT gate is an electronic circuit that produces an inverted version of the input as its output. It is also known as an inverter.

The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high.

The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high.