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Math  /  Discrete

Question5. Construct the truth table for: 6 points each a) ∼p∨q\sim p \vee q \begin{tabular}{|l|l|l|l|} \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & \\ \hline & & \\ \hline \end{tabular} b) ∼(p∨q)\sim(p \vee q) \begin{tabular}{|l|l|l|l|} \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. A truth table is a mathematical table used in logic to compute the functional values of logical expressions on each of their functional arguments.
2. The logical operators involved are negation (∼\sim) and disjunction (∨\vee).
3. The variables pp and qq can each be either true (T) or false (F).

STEP 2

1. Construct the truth table for ∼p∨q\sim p \vee q.
2. Construct the truth table for ∼(p∨q)\sim(p \vee q).

STEP 3

List all possible truth values for pp and qq. Since each can be true or false, there are 4 combinations:
1. p=T,q=Tp = \text{T}, q = \text{T}
2. p=T,q=Fp = \text{T}, q = \text{F}
3. p=F,q=Tp = \text{F}, q = \text{T}
4. p=F,q=Fp = \text{F}, q = \text{F}

STEP 4

Calculate ∼p\sim p for each combination:
1. If p=Tp = \text{T}, then ∼p=F\sim p = \text{F}.
2. If p=Fp = \text{F}, then ∼p=T\sim p = \text{T}.

STEP 5

Calculate ∼p∨q\sim p \vee q for each combination:
1. ∼p=F,q=T⇒∼p∨q=T\sim p = \text{F}, q = \text{T} \Rightarrow \sim p \vee q = \text{T}
2. ∼p=F,q=F⇒∼p∨q=F\sim p = \text{F}, q = \text{F} \Rightarrow \sim p \vee q = \text{F}
3. ∼p=T,q=T⇒∼p∨q=T\sim p = \text{T}, q = \text{T} \Rightarrow \sim p \vee q = \text{T}
4. ∼p=T,q=F⇒∼p∨q=T\sim p = \text{T}, q = \text{F} \Rightarrow \sim p \vee q = \text{T}

STEP 6

Construct the truth table for ∼p∨q\sim p \vee q:
pq∼p∼p∨qTTFTTFFFFTTTFFTT\begin{array}{|c|c|c|c|} \hline p & q & \sim p & \sim p \vee q \\ \hline \text{T} & \text{T} & \text{F} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} \\ \hline \end{array}

STEP 7

Calculate p∨qp \vee q for each combination:
1. p=T,q=T⇒p∨q=Tp = \text{T}, q = \text{T} \Rightarrow p \vee q = \text{T}
2. p=T,q=F⇒p∨q=Tp = \text{T}, q = \text{F} \Rightarrow p \vee q = \text{T}
3. p=F,q=T⇒p∨q=Tp = \text{F}, q = \text{T} \Rightarrow p \vee q = \text{T}
4. p=F,q=F⇒p∨q=Fp = \text{F}, q = \text{F} \Rightarrow p \vee q = \text{F}

STEP 8

Calculate ∼(p∨q)\sim(p \vee q) for each combination:
1. p∨q=T⇒∼(p∨q)=Fp \vee q = \text{T} \Rightarrow \sim(p \vee q) = \text{F}
2. p∨q=T⇒∼(p∨q)=Fp \vee q = \text{T} \Rightarrow \sim(p \vee q) = \text{F}
3. p∨q=T⇒∼(p∨q)=Fp \vee q = \text{T} \Rightarrow \sim(p \vee q) = \text{F}
4. p∨q=F⇒∼(p∨q)=Tp \vee q = \text{F} \Rightarrow \sim(p \vee q) = \text{T}

STEP 9

Construct the truth table for ∼(p∨q)\sim(p \vee q):
pqp∨q∼(p∨q)TTTFTFTFFTTFFFFT\begin{array}{|c|c|c|c|} \hline p & q & p \vee q & \sim(p \vee q) \\ \hline \text{T} & \text{T} & \text{T} & \text{F} \\ \text{T} & \text{F} & \text{T} & \text{F} \\ \text{F} & \text{T} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{F} & \text{T} \\ \hline \end{array}

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