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Complete the following truth table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$$\\var{a1} \\var{op2} \\var{b1}$$(\\var{a} \\var{op} \\var{b}) \\var{op1} (\\var{a1} \\var{op2} \\var{b1})$$\\var{a2} $$((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2} $
$\\var{p[0]}$$\\var{q[0]}$[[0]][[4]][[8]][[12]][[16]]
$\\var{p[1]}$$\\var{q[1]}$[[1]][[5]][[9]][[13]][[17]]
$\\var{p[2]}$$\\var{q[2]}$[[2]][[6]][[10]][[14]][[18]]
$\\var{p[3]}$$\\var{q[3]}$[[3]][[7]][[11]][[15]][[19]]
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In the following question you are asked to construct a truth table for:

\n

\\[((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2}.\\]

\n

\n

Enter T if true, else enter F.

", "advice": "

First we find the truth table for $\\var{a} \\var{op} \\var{b}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev1[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev1[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev1[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev1[3]}$
\n

Then the truth table for $\\var{a1} \\var{op2} \\var{b1}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a1} \\var{op2} \\var{b1}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev2[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev2[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev2[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev2[3]}$
\n

Putting these together to find $(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$:

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$$\\var{a1} \\var{op2} \\var{b1}$$(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev1[0]}$$\\var{ev2[0]}$$\\var{t_value[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev1[1]}$$\\var{ev2[1]}$$\\var{t_value[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev1[2]}$$\\var{ev2[2]}$$\\var{t_value[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev1[3]}$$\\var{ev2[3]}$$\\var{t_value[3]}$
\n

Next we find the truth table for $\\var{a2}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a2}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev3[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev3[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev3[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev3[3]}$
\n

Putting this all together to obtain the truth table we want:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$$\\var{a2}$$((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2} $
$\\var{p[0]}$$\\var{q[0]}$$\\var{t_value[0]}$$\\var{ev3[0]}$$\\var{final_value[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{t_value[1]}$$\\var{ev3[1]}$$\\var{final_value[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{t_value[2]}$$\\var{ev3[2]}$$\\var{final_value[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{t_value[3]}$$\\var{ev3[3]}$$\\var{final_value[3]}$
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Create a truth table for a logical expression of the form $((a \\operatorname{op1} b) \\operatorname{op2}(c \\operatorname{op3} d))\\operatorname{op4}e $ where each of $a, \\;b,\\;c,\\;d,\\;e$ can be one the Boolean variables $p,\\;q,\\;\\neg p,\\;\\neg q$ and each of $\\operatorname{op1},\\;\\operatorname{op2},\\;\\operatorname{op3},\\;\\operatorname{op4}$ one of $\\lor,\\;\\land,\\;\\to$.

\n

For example: $((q \\lor \\neg p) \\to (p \\land \\neg q)) \\lor \\neg q$

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}