Worked Example

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Example 1 (Slide 13)

So let’s take a look at example one. And the best way to start this is where the output is actually true. So if we examine this line here, this line and this line, we can see that when B and C are true, output is True. When a and c are true, the output is True, or when all three are true, the output is also true. So a really good way to start is just by coloring in these three facts.

Venn Diagram

So when B and C are true, that’s this little square here. Then when a and c are true, that’s this little portion of the Venn diagram. And then when a, b, and c are so when all three are true, And that’s this little center part of the diagram. And then all other rows of the truth table are all false or zero. So we’re not going to fill in anything else in the Venn diagram here. So this is the full output of or the full expression represented as a Venn diagram.

Expression

So let’s tackle this as a Boolean expression. And so we’ve already talked through parts of what the Boolean expression would actually be. So this can start here with a and c, when a and c are true, the statement is true. So let’s write a and see us as one portion of our Boolean expression. So I’m going to write that using parentheses, and then we have B and C. So I’m going to kind of write this over here, B and C. What that’s the other part of the truth in our Boolean expression, but we can’t just write them side by side, right, we need to join the middle because we also need the center portion. So it’s either a and c are true, or either A and C or B and C. So enter a or operate in between. So if A and C are true, or B and C are true, so that’s this portion right here. So when a and c, or B and C are true, and our expression is true, this will represent our Boolean logical statement. But there are alternative ways to write this. And these aren’t just the only ways another way you could have wrote this would be C, right? Because in all three cases, C is always true. So C, and a or b.

Logic Gates

So let’s take a look at what the logic gates look like. So the logic gates and I’m going to write the logic gates for my top statement here. So this one right here, we need to write the left and right hand side of the OR operator first, and the OR operator is going to join these two at the end. The lines here represent the inputs of A, B and C. So I’m going to draw out the input of a because in this case, we have a and c. And those are connected together with the AND operator. And that gives an output and we have the same thing with B and C. So we have B, C, those are both connected together with an AND operator and both of those are joined together using or. So this would be the logic gate for this particular statement here.