Logical Equivalence in simple word is the comparision of two logical statements. Meaning of the word logic is analysis. Analysis may be granted result or mathematical proof. Two statemnets satisfies the equivalance if the resultant truth table of both the given statements are exactly same. That is, the resultant truth values are the same. Equivalence statements satisfies the "if and only if" or biconditional operations. If A and B are two equivalant statements then one can be proved from another.

We use '`-=` symbol to represent the term logical equivalence in discrete mathematics. Let us see about logical equivalence in this article.

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If the last column of two given components in the logical equivalence table are the same, then the two components are said to be logical equivalent.

We can use'`-=` ' symbol to represent logical equivalences. Other name of logical equivalence is simple equivalence. These are the logical equivalence laws.

Given below are some of the logical equivalence examples:

**Example 1: **Show ~ (X`vv` Y) = (~X)`^^` (~Y) simple equivalence.

**Proof:**

**LHS:**

~ (X`vv` Y)

X | Y | X`vv` Y | ~ (X`vv` Y) |

T | T | T | F |

T | F | T | F |

F | T | T | F |

F | F | F | T |

**RHS:** ** **

(~X)`^^` (~Y)

X | Y | ~X | ~Y | (~X)`^^` (~Y) |

T | T | F | F | F |

T | F | F | T | F |

F | T | T | F | F |

F | F | T | T | T |

Last column of LHS table is equal to last column of RHS table. And so, it is a simple equivalence.

**Example 2: **Show (X`^^` Y) = ~ ((~X)`vv` (~Y)) simple equivalence.

**Proof:**

**LHS:**

(X`^^` Y)

X | Y | (X`^^` Y) |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

**RHS: **

~ ((~X)`vv` (~Y))

X | Y | ~X | ~Y | ((~X)`vv` (~Y)) | ~ ((~X)`vv` (~Y)) |

T | T | F | F | F | T |

T | F | F | T | T | F |

F | T | T | F | T | F |

F | F | T | T | T | F |

Last column of LHS table is equal to last column of RHS table. And so, it is a simple equivalence.

**Example 3: **Show (X`vv` Y)`vv` (~(X`vv` Y)) =(X`vv` (~Y))`vv` Y simple equivalence.

**Proof:**

**LHS:**

(X`vv` Y)`vv` (~(X`vv` Y))

X | Y | X`vv` Y | (~(X`vv` Y) | (X`vv` Y)`vv` (~(X`vv` Y)) |

T | T | T | F | T |

T | F | T | F | T |

F | T | T | F | T |

F | F | F | T | T |

**RHS: **

(X`vv` (~Y))`vv` Y

X | Y | ~Y | (X`vv` (~Y)) | (X`vv` (~Y))`vv` Y |

T | T | F | T | T |

T | F | T | T | T |

F | T | F | F | T |

F | F | T | T | T |

Last column of LHS table is equal to last column of RHS table. And so, it is a simple equivalence.

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