Subsection 1.1 Negating and Combining Statements with "and" and "or"
Subsubsection 1.1.1 Negations
The negation of a given statement switches the truth value of the original statement. For example, the negation of "Abortion is a federal crime" is "Abortion is not a federal crime". Many times, a statement can be negated by inserting the word "not", or an equivalent word into the original statement. There are many ways to symbolically represent the negation of a statement. For this module, if we represent a statement by the variable \(p\text{,}\) we represent the negation of \(p\) by \(\neg p\text{.}\) We can use a logic calculator to see how the value of \(p\) and \(\neg p\) are related.
In the calculation above, the left most column represents the values for \(p\) and the column under "value" represents the corresponding values for \(\neg p\text{.}\)Subsubsection 1.1.2 And & Or
As we can see, when \(p\) is true, \(~p\) is false and vice versa.The calculator used above generated a truth table. A truth table is a table that will produce the truth values for compound statements based off of the statement variable which are typically placed to the left of the table and the truth values for the compound statement which are typically to the right in the table. We will use this truth table generator for the remainder of this module.
Two more common ways to make a compound statement is by taking two simple statements and combining them with either the word "or" or "and". For example, consider the two statements " In August 2022, abortion is legal with not gestational limit in the state of New Jersey"and "In August 2022, abortion was banned with no exception for rape or incest in the state of Alabama"[1.12.2]. We can combine these statements in the following two ways:
In August 2022, abortion is legal with no gestational limit in the state of New Jersey or in August 2022, abortion was banned with no exception for rape or incest in the state of Alabama.
In August 2022, abortion is legal with no gestational limit in the state of New Jersey and in August 2022, abortion was banned with no exception for rape or incest in the state of Alabama.
These statements are a bit awkward and can be rewritten without losing their meaning as simply:
In August 2022, abortion is legal with not gestational limit in the state of New Jersey or was banned with no exception for rape or incest in the state of Alabama.
In August 2022, abortion is legal with not gestational limit in the state of New Jersey and was banned with no exception for rape or incest in the state of Alabama.
Both of these compound statements are true statements because the original simple statements are both true. The truth table for an "or" statement, more formally known as a disjunction is given below:
The truth table for an "and" statement, more formally known as a conjunction is given next: The columns on the far left represent all possible truth values of the simple statements \(p\) and \(q\) used in the compound statement. The entries on the far left of each row represent the different truth value of the simple statements used in the compound statement. The final column on the far right represents the truth value of the compound statement for the given simple statement truth values. For example the second row in the truth table above reads "When \(p\) is true and \(q\) is false, \(p\) and \(q\) is false".