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The symbol V is sometimes called the inclusive or because p V q is true when p is true or when q is true, or when both are true. That is, we include the case when both statements are true. On the other hand, the exclusive or operator © is defined so that p ℗ q is true exactly when one (but not both) of p and q are true. That is, we exclude the case when both p and q are true. In natural conversation you might hear the phrase "either p or q" for p q. (a) Write down the truth table for . (b) It turns out that the set of connectives {^, V,¬} is what we can call a basis. This means that every truth table in two variables can be generated by these (and no other) connectives. A statement written using only these connectives is said to be in disjunctive normal form, as long as the symbol is only used directly before a propositional variable. 7 Write down a logically equivalent statement to p q which is in disjunctive normal form. Prove your statement is equivalent by writing out its truth table in an organized manner. (c) For each sub-task, determine whether the given set B of operations forms a basis. If it does, give a brief justification. If it does not, give at least one truth table in variables x, y for which there is no compound proposition using the operations from B that has exactly that truth table. i) B = {^,¬} ii) B = {V,¬} iii) B = {V, ^} iv) B = {→,¬}

The symbol V is sometimes called the inclusive or because p V q is true when p is true or when q is true, or when both are true. That is, we include the case when both statements are true. On the other hand, the exclusive or operator © is defined so that p ℗ q is true exactly when one (but not both) of p and q are true. That is, we exclude the case when both p and q are true. In natural conversation you might hear the phrase "either p or q" for p q. (a) Write down the truth table for . (b) It turns out that the set of connectives {^, V,¬} is what we can call a basis. This means that every truth table in two variables can be generated by these (and no other) connectives. A statement written using only these connectives is said to be in disjunctive normal form, as long as the symbol is only used directly before a propositional variable. 7 Write down a logically equivalent statement to p q which is in disjunctive normal form. Prove your statement is equivalent by writing out its truth table in an organized manner. (c) For each sub-task, determine whether the given set B of operations forms a basis. If it does, give a brief justification. If it does not, give at least one truth table in variables x, y for which there is no compound proposition using the operations from B that has exactly that truth table. i) B = {^,¬} ii) B = {V,¬} iii) B = {V, ^} iv) B = {→,¬}

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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please only answer part c

The symbol V is sometimes called the inclusive or because p V q is true when p is true or when q is true, or when both are true. That is, we include the case when both statements are true. On the other hand, the exclusive or operatoris defined so that p q is true exactly when one (but not both) of p and q are true. That is, we exclude the case when both and q are true. In natural conversation you might hear the phrase "either p or q" for pq. р (a) Write down the truth table for . (b) It turns out that the set of connectives {^, V,¬} is what we can call a basis. This means that every truth table in two variables can be generated by these (and no other) connectives. A statement written using only these connectives is said to be in disjunctive normal form, as long as the symbol is only used directly before a propositional variable. Write down a logically equivalent statement to p q which is in disjunctive normal form. Prove your statement is equivalent by writing out its truth table in an organized manner. (c) For each sub-task, determine whether the given set B of operations forms a basis. If it does, give a brief justification. If it does not, give at least one truth table in variables x, y for which there is no compound proposition using the operations from B that has exactly that truth table. i) B = {^, ¬} ii) B = {V,¬} iii) B = {V, ^} iv) B = {→→}
Transcribed Image Text:The symbol V is sometimes called the inclusive or because p V q is true when p is true or when q is true, or when both are true. That is, we include the case when both statements are true. On the other hand, the exclusive or operatoris defined so that p q is true exactly when one (but not both) of p and q are true. That is, we exclude the case when both and q are true. In natural conversation you might hear the phrase "either p or q" for pq. р (a) Write down the truth table for . (b) It turns out that the set of connectives {^, V,¬} is what we can call a basis. This means that every truth table in two variables can be generated by these (and no other) connectives. A statement written using only these connectives is said to be in disjunctive normal form, as long as the symbol is only used directly before a propositional variable. Write down a logically equivalent statement to p q which is in disjunctive normal form. Prove your statement is equivalent by writing out its truth table in an organized manner. (c) For each sub-task, determine whether the given set B of operations forms a basis. If it does, give a brief justification. If it does not, give at least one truth table in variables x, y for which there is no compound proposition using the operations from B that has exactly that truth table. i) B = {^, ¬} ii) B = {V,¬} iii) B = {V, ^} iv) B = {→→}
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