2.6 De Morgan’s Laws

1. Kristin is a biomedical engineer and Thomas is a chemical engineer has the form “$p\wedge q$,” where $p$ is the statement, “Kristin is a biomedical engineer,” and $q$ is the statement, “Thomas is a chemical engineer.” By De Morgan’s Law, the negation of a conjunction, $~(p\wedge q)$, is logically equivalent to $~p\vee ~q.$ $~p$ is “Kristen is not a biomedical engineer,” and $~q$ is “Thomas is not a chemical engineer.” By De Morgan’s Law, the solution has the form “$~p\vee ~q$,” so the answer is: “Kristin is not a biomedical engineer or Tom is not a chemical engineer.”
2. A person had cake or they had ice cream has the form “$p\vee q,$” where $p$ is the statement, “A person had cake,” and $q$ is the statement, “A person had ice cream.” By De Morgan’s Law for the negations of a disjunction, $~(p\vee q)\equiv ~p\wedge ~q.$ The solution is the statement: “A person did not have cake and they did not have ice cream.”

1. The negation of the conditional statement, $p\to q,$ is the statement, $p\wedge ~q.$ The hypothesis of the conditional statement is $p$: “Adele won a Grammy,” and conclusion of the conditional statement is $q$: “Adele is a singer.” The negation of the conclusion, $~q$, is the statement: “She is not a singer.” Therefore, the answer is $p\wedge ~q:$ “Adele won a Grammy, and she is not a singer.”
2. The hypothesis is $p$: “Henrik Lundqvist played professional hockey,” and the conclusion of the conditional statement is $q$: “He did not win the Stanley Cup.” The negation of $q$ is the statement: “He won the Stanley Cup.” The negation of the conditional statement is equal to $p\wedge ~q:$ “Henrick Lundqvist played professional hockey, and he won the Stanley Cup.”

1. The negation of the conditional statement $p\to q$ is the statement $p\wedge ~q.$ The hypothesis of the conditional statement is $p$: “All cats purr,” and the conclusion of the conditional statement is $q$: “My partner’s cat purrs.” The negation of the conclusion, $~q$, is the statement: “My partner’s cat does not purr.” Therefore, the answer is $p\wedge ~q:$ “All cats purr, but my partner’s cat does not purr.”
2. The hypothesis is $p$: “A penguin is a bird,” and the conclusion of the conditional statement is $q$: “Some birds do not fly.” The negation of $q$ is the statement: “All birds fly.” Therefore, the negation of the conditional statement is equal to $p\wedge ~q:$ “A penguin is a bird, and all birds fly.”

1. The conditional has the form “If $p$ then $q$ or $r$,” where $p$ is “Mom needs to buy chips,” $q$ is “Mike had friends over,” and $r$ is “Bob was hungry.” The negation of $p\to (q\wedge r)$ is $p\wedge ~(q\wedge r).$ Applying De Morgan’s Law to the statement $~(q\wedge r)$ the result is $~q\vee ~r$, so our conditional statement becomes $p\wedge (~q\vee ~r).$ By the distributive property for conjunction over disjunction, this statement is equivalent to $\left(p\wedge ~q\right)\vee \left(p\wedge ~r\right).$ Translating the statement $\left(p\wedge ~q\right)\vee \left(p\wedge ~r\right)$ into words, the solution is: “Mom needs to buy chips and Mike did not have friends over, or Mom needs to buy chips and Bob was not hungry.”
2. The conditional has the form “If $p$ or $q$, then $r$,” where $p$ is “Juan had pizza,” $q$ is “Chris had wings,” and $r$ is “Dad watched the game.” The negation of $\left(p\vee q\right)\to r$ is $\left(p\vee q\right)\wedge ~r.$ By the distributive property for disjunction over conjuction, the statement is equivalent to $\left(p\vee ~r\right)\wedge \left(q\vee ~r\right).$ Translating the statement $\left(p\vee ~r\right)\wedge \left(q\vee ~r\right)$ into words, the solution is: “Juan had pizza or dad did not watch the game, and Chris had wings or dad did not watch the game.”

Step 1: To verify any logical equivalence, you must first replace the logical equivalence symbol, $\equiv$, with the biconditional symbol, $\leftrightarrow$. The statement $~\left(p\wedge q\right)\equiv ~p\vee ~q$ becomes $~\left(p\wedge q\right)\leftrightarrow ~p\vee ~q.$
Step 2: Next, you create a truth table for the statement. Because we have two basic statements, $p$, and $q$, the truth table will have four rows to account for all the possible outcomes. The columns will be $p$, $q$, $~p$, $~q$, $p\wedge q,$$~(p\wedge q),$$~p\vee ~q,$ and the biconditional statement is $~\left(p\wedge q\right)\leftrightarrow ~p\vee ~q.$

Step 3: Finally, verify that the statement is valid by confirming it is a tautology. In this instance, the last column is all true. Therefore, the statement is valid and De Morgan’s Law for the negation of a conjunction is verified.