Proving the Associative Property of Logical AND and Logical OR: Understanding the Truth Table and Evaluating Two Given Expressions

p ∧ (q ∧ r ) ≡ (p ∧ q) ∧ rp ∨ (q ∨ r ) ≡ (p ∨ q) ∨ r

To prove the two given expressions:

Expression 1: p ∧ (q ∧ r ) ≡ (p ∧ q) ∧ r

Using the associative property of ∧ (logical AND), we can rearrange the parentheses:

p ∧ (q ∧ r ) ≡ (p ∧ r) ∧ q

Now let’s consider the truth table for logical AND:

p | q | r | q ∧ r | p ∧ (q ∧ r) | (p ∧ r) ∧ q
—————————————–
T | T | T | T | T | T
T | T | F | F | F | F
T | F | T | F | F | F
T | F | F | F | F | F
F | T | T | T | F | F
F | T | F | F | F | F
F | F | T | F | F | F
F | F | F | F | F | F

As we can see from the truth table, the values for both Expression 1 are always the same (either T or F) for all possible combinations of p, q, and r

To prove the two given expressions:

Expression 1: p ∧ (q ∧ r ) ≡ (p ∧ q) ∧ r

Using the associative property of ∧ (logical AND), we can rearrange the parentheses:

p ∧ (q ∧ r ) ≡ (p ∧ r) ∧ q

Now let’s consider the truth table for logical AND:

p | q | r | q ∧ r | p ∧ (q ∧ r) | (p ∧ r) ∧ q
—————————————–
T | T | T | T | T | T
T | T | F | F | F | F
T | F | T | F | F | F
T | F | F | F | F | F
F | T | T | T | F | F
F | T | F | F | F | F
F | F | T | F | F | F
F | F | F | F | F | F

As we can see from the truth table, the values for both Expression 1 are always the same (either T or F) for all possible combinations of p, q, and r. Therefore, we can conclude that Expression 1 is true.

Expression 2: p ∨ (q ∨ r ) ≡ (p ∨ q) ∨ r

Using the associative property of ∨ (logical OR), we can rearrange the parentheses:

p ∨ (q ∨ r ) ≡ (p ∨ r) ∨ q

Now let’s consider the truth table for logical OR:

p | q | r | q ∨ r | p ∨ (q ∨ r) | (p ∨ r) ∨ q
—————————————–
T | T | T | T | T | T
T | T | F | T | T | T
T | F | T | T | T | T
T | F | F | F | T | T
F | T | T | T | T | T
F | T | F | T | T | T
F | F | T | T | T | T
F | F | F | F | F | F

As we can see from the truth table, the values for both Expression 2 are always the same (either T or F) for all possible combinations of p, q, and r. Therefore, we can conclude that Expression 2 is true.

So, both Expression 1 and Expression 2 are true.

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