Compound Sentences
A compound sentence is formed when two or more thoughts are connected in one sentence.
The following are examples of compound sentences:
“21 is divisible by 3 and 21 is not prime.”
“45 is a multiple of 9 or 13 – 20 = 7.”
“If 4 + 6 = 10 and 3 + 3 = 9, then all rectangles are squares.”
When attempting to determine the truth value of a compound sentence, first determine the truth value of each of the components of the sentence.
Let’s examine the examples listed above.
- Determine the truth value of: “21 is divisible by 3 and 21 is not prime.”
“21 is divisible by 3” (true)
“21 is not prime” (true)
Substitute the truth values for the facts: T and T
Simplify the conjunction (and): T
Answer: The compound sentence (statement) is true. - Determine the truth value of: “45 is a multiple of 9 or 13 – 20 = 7.”
“45 is a multiple of 9” (true)
“13 – 20 = 7” (false)
Substitute the truth values for the facts: T or F
Simplify the disjunction (or): T
Answer: The compound sentence (statement) is true. - Determine the truth value of:
“If 4 + 6 = 10 and 3 + 3 = 9, then all rectangles are squares.”
“4 + 6 = 10 ” (true)
“3 + 3 = 9” (false)
“all rectangles are squares.” (false)
Substitute the truth values for the facts: if (T and F) then F
Simplify the conjunction (and) first: if F then F
Simplify the conditional: T
Answer: The compound sentence (statement) is true.
When the truth value of one or more of the components of a compound sentence is unknown, all of the possible truth values must be considered. A truth table is the easiest way to show all of these possibilities.
Construct a truth table for (p ∧ q) →∼ (p ∨ q)
| p | q | p ∧ q | p ∨ q | ∼ (p ∨ q) | (p ∧ q) →∼ (p ∨ q) |
| T | T | T | T | F | F |
| T | F | F | T | F | T |
| F | T | F | T | F | T |
| F | F | F | F | T | T |
The truth table tells you that the compound sentence will be false only when p and q are both true. In all other situations, the compound sentence is true.