Question

Which of the following is true ? (a) \(\mathrm{p} \Lambda(\sim \mathrm{p})=\mathrm{t}\) (b) \(p V(\sim p)=f\) (c) \(p \Rightarrow q=q \Rightarrow p\) (d) \(\mathrm{p} \Rightarrow \mathrm{q}=(\sim \mathrm{q}) \Rightarrow(\sim \mathrm{p})\)

Short answer

The correct answer is (d) \(\mathrm{p} \Rightarrow \mathrm{q}=(\sim \mathrm{q}) \Rightarrow(\sim \mathrm{p})\).

Step-by-step solution

Statement (a)

We are given that \(\mathrm{p} \Lambda(\sim \mathrm{p})=\mathrm{t}\), and we will need to check if it's true. \(\mathrm{p} \Lambda(\sim \mathrm{p})\) means \(\mathrm{p}\) AND NOT \(\mathrm{p}\). The truth table for AND is as follows: | p | NOT p | p AND NOT p | |---|-------|-------------| | T | F | F | | F | T | F | As we can see, both combinations of p (T or F) give us F for the compound statement \(\mathrm{p} \Lambda(\sim \mathrm{p})\). Thus, the statement (a) is false, as it is not equal to t.

Statement (b)

We are given that \(p V(\sim p)=f\), and we will need to check if it's true. \(p V(\sim p)\) means \(p\) OR NOT \(p\). The truth table for OR is as follows: | p | NOT p | p OR NOT p | |---|-------|------------| | T | F | T | | F | T | T | As we can see, both combinations of p (T or F) give us T for the compound statement \(p V(\sim p)\). Thus, the statement (b) is false, as it is not equal to f.

Statement (c)

We are given that \(p \Rightarrow q=q \Rightarrow p\), and we need to check if they are equivalent. \(p \Rightarrow q\) means IF \(p\), THEN \(q\). The truth table for IMPLIES is as follows: | p | q | p IMPLIES q | q IMPLIES p | |---|---|-------------|-------------| | T | T | T | T | | T | F | F | T | | F | T | T | F | | F | F | T | T | As we can see, in two rows (second row and third row) the results for \(p \Rightarrow q\) and \(q \Rightarrow p\) are different. Thus, the statement (c) is false, as \(p \Rightarrow q\) is not equivalent to \(q \Rightarrow p\).

Statement (d)

We are given that $\mathrm{p} \Rightarrow \mathrm{q}=(\sim \mathrm{q}) \Rightarrow(\sim \mathrm{p})$, and we will need to check if they are equivalent. The truth table is as follows: | p | q | NOT q | NOT p | p IMPLIES q | NOT q IMPLIES NOT p | |---|---|-------|-------|-------------|---------------------| | T | T | F | F | T | T | | T | F | T | F | F | F | | F | T | F | T | T | T | | F | F | T | T | T | T | As we can see, for all combinations of p and q, the results for \(\mathrm{p} \Rightarrow \mathrm{q}\) and \((\sim \mathrm{q}) \Rightarrow(\sim \mathrm{p})\) are equal. Thus, statement (d) is true. The correct answer is: (d) \(\mathrm{p} \Rightarrow \mathrm{q}=(\sim \mathrm{q}) \Rightarrow(\sim \mathrm{p})\)

Key concepts

Truth Tables

Truth tables are a wonderful tool in the realm of propositional logic. They help us keep track of the truth values of different logical expressions. Imagine them like a map that shows each possible state of your variables and what those states lead to for your logical statement.
To construct a truth table, you make a row for every possible combination of truth values for your variables (let's say they're true or false, T or F). You then follow each logical operation one by one and record what truth value they produce in those specific cases. This systematic method allows you to analyze any logical expression thoroughly.
For example, if you have two variables, like \(p\) and \(q\), you'll need four combinations: both true, both false, one true and the other false, and vice versa. Then you'll compute the outcome for each possible combination using logical connectives such as AND (\(\land\)), OR (\(\lor\)), NOT (\(\sim\)), IMPLIES (\(\Rightarrow\)), etc.

• The AND operation is true only if both operands are true.
• The OR operation is true if at least one operand is true.
• The NOT operation simply flips the truth value.

Truth tables are critical in making logical reasoning concrete, helping verify properties like equivalence or predicting results of complex expressions.

Logical Equivalence

Logical equivalence is a concept that helps us understand when two statements can be considered logically the same. Two statements are logically equivalent if they have exactly the same truth value across all possible interpretations. In means whenever one is true, the other is true as well, and similarly for false. They behave identically no matter how the underlying variables are set.
To determine equivalence, you can build a truth table for each statement and compare the results for all variable combinations. If for every possible input the outputs are identical, then the two statements are logically equivalent. For instance, "NOT \(p\) OR \(q\)" is equivalent to "\(p\) IMPLIES \(q\)". This kind of transformation is essential in logic because it allows simplification and transformation of complex expressions into simpler forms.
Logical equivalence allows one to replace one statement with another, simpler, equally valid one. This process can significantly ease solving logical problems and understanding how different logical expressions are interconnected.

Logical Connectives

Logical connectives are the building blocks of propositional logic. They are the symbols or words used to combine simpler statements into more complex statements. Imagine them as the glue holding our logical statements together, helping us form nuanced and sophisticated ideas from basic expressions.
The most common logical connectives include:

• AND (\(\land\)) - True if both statements are true.
• OR (\(\lor\)) - True if at least one statement is true.
• NOT (\(\sim\)) - Inverts the truth value.
• IMPLIES (\(\Rightarrow\)) - Represents "if...then...". It's false only when the first statement is true, and the second is false.
• BICONDITIONAL (\(\Leftrightarrow\)) - True if both statements are either true or false.

These connectives allow you to articulate conditions and relations between statements. Mastering them is key to constructing and understanding complex logical arguments, as each connective has its unique way of altering or combining the truth values of statements.