a. (p∧q)∨(p∨r)=p∨(q∧r)
Solution.
pFqFrTp∧qFp∨rT(p∧q)∨(p∨r)Tq∧rFp∨(q∧r)F There is a row in the truth table above where the truth values of (p∧q)∨(p∨r) and p∨(q∧r) are different. Hence the given statement is false. (In this case, other rows of the truth table are not important.)
Answer. False.
b. (p∨q)∧(p∨r)=p∨(q∧r)
Solution.
pFFFFTTTTqFFTTFFTTrFTFTFTFTp∨qFFTTTTTTp∨rFTFTTTTT(p∨q)∧(p∨r)FFFTTTTTq∧rFFFTFFFTp∨(q∧r)FFFTTTTT In all rows, the truth values of (p∨q)∧(p∨r) and p∨(q∧r) are equal. Hence the given statement is true.
Answer. True.
c. ∼(p∨q)=∼(p∧∼q)
Solution.
pFqTp∨qT∼(p∨q)F∼qFp∧∼qF∼(p∧∼q)T There is a row in the truth table above where the truth values of ∼(p∨q) and ∼(p∧∼q) are different. Hence the given statement is false. (In this case, other rows of the truth table are not important.)
Answer. False.
d. (p∧q)∧(p∨r)=p∨(q∧r)
Solution.
pTqFrFp∧qFp∨rT(p∧q)∧(p∨r)Fq∧rFp∨(q∧r)T There is a row in the truth table above where the truth values of (p∧q)∧(p∨r) and p∨(q∧r) are different. Hence the given statement is false. (In this case, other rows of the truth table are not important.)
Answer. False.
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