Logic, Conditionals & Venn Diagrams

What you'll learn

Identify the hypothesis and conclusion of a conditional statement
Write the converse, inverse, and contrapositive of a conditional
Use Venn diagrams to show relationships between sets

Words to know

Conditional —: an if-then statement: if p, then q
Hypothesis —: the 'if' part (p) of a conditional
Conclusion —: the 'then' part (q) of a conditional
Converse —: swap the parts: if q, then p
Contrapositive —: negate and swap: if not q, then not p
Venn diagram —: overlapping regions that show how sets relate

Why it matters Clear logic lets us state a fact, test whether its reverse is also true, and picture how categories fit inside one another.

Worked examples

Model ItI do

For 'If a shape is a square, then it is a rectangle,' name the hypothesis and conclusion, and use the Venn diagram to explain why.

Hypothesis: 'a shape is a square.' Conclusion: 'it is a rectangle.' In the Venn diagram the Squares circle sits entirely inside the Rectangles circle, so every square is a rectangle. The conditional is true.

Try It TogetherWe do

Write the converse of 'If an angle is a right angle, then it measures 90 degrees.' Is the converse true?

Swap the parts. Converse: 'If an angle measures 90 degrees, then it is a right angle.' A 90 degree angle is by definition a right angle, so this converse is also true.

Now YouYou do

Write the contrapositive of 'If two lines are parallel, then they do not intersect.'

Negate both parts and swap them. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

1.Name the conclusion in 'If it snows, then school is closed.'
2.The 'if' part of a conditional statement is called the _____.
3.In a Venn diagram, circle A is drawn entirely inside circle B. Is every A also a B?

On Level

The core Quick Check from the book.

1.Name the hypothesis in 'If it rains, then the game is canceled.'
2.Write the converse of 'If a number is even, then it is divisible by 2.'
3.True or false: the converse of a true statement is always true.
4.In a Venn diagram, if circle A is inside circle B, is every A also a B?
5.Write the inverse of 'If a polygon is a triangle, then it has 3 sides.'

Challenge

Stretch your thinking.

1.Write the contrapositive of 'If two lines are parallel, then they do not intersect.'
2.A conditional statement is true. Which related statement is ALWAYS also true?
3.Write the inverse of 'If a figure is a square, then it has four sides.'