Logic and Proof

Logic and proof questions set the CSSE apart from many other 11+ exams. The CSSE maths paper is designed to test your reasoning ability, not just your calculation skills. This lesson covers the logical thinking skills you need: working systematically, using deduction, spotting patterns, and explaining your reasoning.

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What Is Mathematical Logic?

Mathematical logic means using clear, step-by-step reasoning to solve problems. It involves:

• Deduction: Working out something that must be true based on given information.
• Elimination: Ruling out possibilities until you find the answer.
• Proof: Showing that a statement is always true (or giving a counter-example to show it is false).

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True, False, and Sometimes True

Some statements in maths are always true, some are always false, and some are sometimes true.

Statement	Always / Sometimes / Never true
The sum of two even numbers is even	Always true
A square number is always even	Sometimes true (4 is even, 9 is odd)
An odd number plus an even number is even	Never true (odd + even = odd)
A prime number is always odd	Sometimes true (2 is prime and even)

CSSE Tip: When a question says "Is this statement always, sometimes, or never true?", you must give an example to support your answer. A single counter-example is enough to show something is not always true.

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Working Systematically

Many CSSE questions require you to list possibilities in an organised way.

Worked Example: Combinations

A cafe offers 3 types of sandwich (ham, cheese, egg) and 2 drinks (tea, coffee). How many different meal combinations are there?

List them systematically:

Sandwich	Drink	Combination
Ham	Tea	Ham + Tea
Ham	Coffee	Ham + Coffee
Cheese	Tea	Cheese + Tea
Cheese	Coffee	Cheese + Coffee
Egg	Tea	Egg + Tea
Egg	Coffee	Egg + Coffee

Total = 3 × 2 = 6 combinations

Worked Example: Digit Puzzles

Using the digits 1, 2, and 3 (each used exactly once), how many different three-digit numbers can you make?

• 123, 132, 213, 231, 312, 321
• 6 different numbers

The formula for arrangements of n different items is n! (n factorial).

• 3! = 3 × 2 × 1 = 6

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Deduction and Elimination

Worked Example: Number Puzzle

I am thinking of a number between 1 and 30.

• It is even.
• It is a multiple of 3.
• It is greater than 15.

Step 1: Even numbers between 1 and 30: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30

Step 2: Of these, multiples of 3: 6, 12, 18, 24, 30

Step 3: Greater than 15: 18, 24, 30

If additional information narrows it to one answer, that is the solution.

Worked Example: Logic Grid

Ali, Ben, and Cara each have a different pet: a cat, a dog, and a fish.

• Ali does not have a dog.
• Ben does not have a cat.
• Cara has a fish.

	Cat	Dog	Fish
Ali	?	No	?
Ben	No	?	?
Cara	No	No	Yes

Since Cara has a fish, nobody else does. Ali does not have a dog, so Ali has a cat. That leaves Ben with a dog.

Person	Pet
Ali	Cat
Ben	Dog
Cara	Fish

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Explaining Your Reasoning

The CSSE values clear explanations. When a question says "Explain how you know" or "Show that...", you must write a logical argument.

Worked Example

Show that the sum of any three consecutive numbers is always a multiple of 3.

Let the three consecutive numbers be n, n + 1, n + 2.

• Sum = n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
• Since 3(n + 1) is 3 × a whole number, it is always a multiple of 3.

Worked Example

Is 91 a prime number? Explain your answer.