## Mathematical Reasoning

# Special Words/Phrases

- The compound statement with 'And' is true if all its component statements are true.
- The compound statement with 'And' is false if any of its component statements is false (this includes that case that some of its component statements are false or all its component statements are false).
- A compound statement with an 'Or' is true when one component statement is true or both the component statements are true.
- A compound statement with an 'Or' is false when both the component statements are false.
- The phrases "for all", "for some", "for no", "for every" and "there exists at least one" convey the idea of quantity and refer to some specific collection of number of objects. These phrases quantify the variables in open sentences. They are called quantifiers.
- The quantifier "for all" (or) "for every" are called the "universal quantifiers".
- The quantifiers "some" (or) "there exists at least one" are called "existential quantifiers"
- Dual of the statement : Two compound statement S
_{1}and S_{2}are said to be dual to each other, if one can be obtained by the other by replacing ∧ by ∨ and ∨ by ∧ and t by c and c by t. - Logically equivalence : Two statement S
_{1}and S_{2}are said to be logically equivalence, if they have the same entries in the last column of the truth tables.

Eg:- p → q ⇒ ∼p ∨ q - In the switching circuits applications, the series combination is represented by ∧
- In the switching circuit application, the parallel combination is represented by ∨

### Connectives and or

### Exclusive and Inclusive or

### Quantifiers

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1.** Truth table**

p | q | p ∧ q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

2.

p | q | p ∨ q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |