Answer to Question #94999 in Discrete Mathematics for Tijjani

Question #94999
which of the following statement is true

a. \\((p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\\)
b. \\((p\\vee q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)
c. \\(\\sim (p\\vee q)=\\sim (p\\wedge \\sim q)\\)
d. \\((p\\wedge q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)
1
Expert's answer
2019-09-23T13:15:41-0400

a. "(p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)"

Solution.

"\\begin{matrix}\n\tp & q & r\n\t& p\\wedge q & p\\vee r & (p\\wedge q)\\vee (p\\vee r)\n\t& q\\wedge r & p\\vee (q\\wedge r)\n\t\\\\ F & F & T & F & T & T & F & F\n\\end{matrix}"

There is a row in the truth table above where the truth values of "(p\\wedge q)\\vee (p\\vee r)"  and "p\\vee (q\\wedge 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\\vee q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)"

Solution.

"\\begin{matrix}\n\tp & q & r\n\t& p\\vee q & p\\vee r & (p\\vee q)\\wedge (p\\vee r)\n\t& q\\wedge r & p\\vee (q\\wedge r)\n\t\\\\ F & F & F & F & F & F & F & F\n\t\\\\ F & F & T & F & T & F & F & F\n\t\\\\ F & T & F & T & F & F & F & F\n\t\\\\ F & T & T & T & T & T & T & T\n\t\\\\ T & F & F & T & T & T & F & T\n\t\\\\ T & F & T & T & T & T & F & T\n\t\\\\ T & T & F & T & T & T & F & T\n\t\\\\ T & T & T & T & T & T & T & T\n\\end{matrix}"

In all rows, the truth values of "(p\\vee q)\\wedge (p\\vee r)"  and "p\\vee (q\\wedge r)"  are equal. Hence the given statement is true.

Answer. True.

c. "\\sim (p\\vee q) = \\sim (p\\wedge \\sim q)"

Solution.

"\\begin{matrix}\n\tp & q\n\t& p\\vee q & \\sim (p\\vee q)\n\t& \\sim q & p\\wedge \\sim q & \\sim (p\\wedge \\sim q)\n\t\\\\ F & T & T & F & F & F & T\n\\end{matrix}"

There is a row in the truth table above where the truth values of "\\sim (p\\vee q)"  and "\\sim (p\\wedge \\sim 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\\wedge q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)"

Solution.

"\\begin{matrix}\n\tp & q & r\n\t& p\\wedge q & p\\vee r & (p\\wedge q)\\wedge (p\\vee r)\n\t& q\\wedge r & p\\vee (q\\wedge r)\n\t\\\\ T & F & F & F & T & F & F & T\n\\end{matrix}"

There is a row in the truth table above where the truth values of "(p\\wedge q)\\wedge (p\\vee r)"  and "p\\vee (q\\wedge 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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