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Magazine Problem 56
Construct a truth table for each proposition. $$(p \wedge q) \rightarrow(p \vee q)$$
Problem 56
A third useful quantifier is the uniqueness quantifier \(\exists ! .\) The proposition \(\left(\exists^{\prime} x\right) P(x)\) means There exists a unique (meaning exactly one) \(x\) such that \(P(x) .\) Determine the truth value of each proposition, where UD \(=\) set of integers. $$\left(\exists^{\prime} x\right)(x+3=3)$$
Problem 57
Construct a truth table for each proposition. $$(p \vee q) \leftrightarrow(p \wedge q)$$
Problem 58
Determine whether or not each is a tautology. $$p \vee(\sim p)$$
Problem 58
The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p | q \equiv \sim(p \wedge q)$$
Problem 59
The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p \wedge q \equiv(p | q)|(p | q)$$
Problem 59
Determine whether or not each is a tautology. $$[p \wedge(p \rightarrow q)] \rightarrow q$$
Problem 60
Determine whether or not each is a tautology. $$[(p \rightarrow q) \wedge(\sim q)] \rightarrow \sim p$$
Problem 61
The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p \rightarrow q \equiv((p | p)|(p | p))|(q | q)$$
Problem 61
A third useful quantifier is the uniqueness quantifier \(\exists ! .\) The proposition \(\left(\exists^{\prime} x\right) P(x)\) means There exists a unique (meaning exactly one) \(x\) such that \(P(x) .\) Determine the truth value of each proposition, where UD \(=\) set of integers. $$(\forall x)(\exists ! y)(x+y=4)$$
